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Hmericau  /IDatbematical  Series 

E.   J.   TOWNSEND 

GENERAL   EDITOR 


MATHEMATICAL  SERIES 

E.  J.  TOWNSEND,  General  Editor. 

While  this  series  has  been  planned  to  meet  the  needs  of  the  student  who 
Js  preparing  for  engineering  work,  it  is  hoped  that  it  will  serve  equally  well 
the  purposes  of  those  schools  where  mathematics  is  taken  as  an  element  in 
a  liberal  education.  In  order  that  the  applications  introduced  may  be  of 
such  character  as  to  interest  the  general  student  and  to  train  the  prospective 
engineer  in  the  kind  of  work  which  he  is  most  likely  to  meet,  it  has  been 
the  policy  of  the  editors  to  select,  as  joint  authors  of  each  text,  a  mathemati- 
cian and  a  trained  engineer  or  physicist. 

The  following  texts  are  ready : 

L  Calculus, 

By  E.  J.  TowNSEND,  Professor  of  Mathematics,  and  G.  A.  GooD- 
ENOUGH,  Professor  of  Thermodynamics,  University  of  Illinois.     ^2.50. 

IL  Essentials  of  Calcultis. 

By  E.  J.  TOWNSEND  and  G.  A.  Goodenough.  ^2.00. 

IIL  CoIIegfe  Algebra. 

By  H.  L.  RiETZ,  Assistant  Professor  of  Mathematics,  and  Dr.  A.  R. 
CraTHORNE,  Associate  in  Mathematics  in  the  University  of  Illinois. 
$1.40. 

IV,   Plane  Trigfonometry,  with   Trigfonometric  and 
Logfarithmic  Tables. 

By  A.  G.  Hall,  Professor  of  Mathematics  in  the  University  of  Michigan, 
and  F.  H.  Frink,  Professor  of  Railway  Engineering  in  the  University 
of  Oregon.     $1.25. 

V.  Plane  and  Spherical  Trig-onometry. 

(Without  Tables) 
By  A.  G.  Hall  and  F.  H.  Frink.    ^i.oo. 

VI.  Trigonometric  and  Logarithmic  Tables. 

By  A.  G.  Hall  and  F.  H.  Frink.    75  cents. 

VII.   Analytic  Geometry  of  Space. 

By  Virgil  Snyder,  Professor  in  Cornell  University,  and  C.  H.  SiSAM, 
Assistant  Professor  in  the  University  of  Illinois.    ^2.50. 

VIIL  Analytic  Geometry. 

By  L.  W.  DowLiNG,  Assistant  Professor  of  Mathematics,  and  F.  E. 
TURNEAURE,  Dean  of  the  College  of  Engineering  in  the  University  of 
Wisconsin. 

IX.  Elementary  Geometry. 

By  J.  W.  Young,  Professor  of  Mathematics  in  Dartmouth  College,  and 
a'.  J.  Schwartz,  William  McKinley  High  School,  St.  Louis. 

HENRY     HOLT    AND     COMPANY 

NEW  YORK  CHICAGO 


ANALYTIC   GEOMETRY 


BY 


L.  WAYLAND   BOWLING,  Ph.D. 

Associate  Professob  of  Mathematics 
University  of  Wisconsin 

AND 

F.  E.  TURNEAURE,  C.E. 

Dean  of  the  College  of  Engineering 
University  of  Wisconsin 


BOSTON  ooLLEOK,„.„ 


^4 

NEW  YORK 
HENRY  HOLT  AND   COMPANY 


^^. 


150563 


Copyright,  1914 

BY 

HENET   HOLT   AND   COMPANY 


NorSiiooli  }Pr£03 

J.  S.  Gushing  Co.  —  Berwick  &  Smith  Co. 

Norwood,  Mass.,  U.S.A. 


PREFACE 

Ix  accordance  with  the  general  plan  of  this  series  of  textbooks, 
the  authors  of  the  present  volume  have  had  constantly  in  mind 
the  needs  of  the  student  who  takes  his  mathematics  primarily 
with  a  view  to  its  applications  as  well  as  the  needs  of  the  student 
who  pursues  mathematics  as  an  element  of  his  education. 

The  processes  of  analytical  geometry  find  their  application,  for 
the  most  part,  in  the  scientific  laboratory  where  it  is  often  neces- 
sary to  study  the  properties  of  a  function  from  certain  observed 
values.  The  fundamental  concept  is,  therefore,  that  of  functional 
correspondence  and  the  methods  of  representing  such  correspond- 
ence geometrically.  For  this  reason  rather  more  than  usual 
attention  has  been  given  to  these  subjects  (Chapter  III;  also 
Chapter  IX,  Arts.  135  to  140). 

An  intelligent  appreciation  of  functional  correspondence  re- 
quires an  intimate  knowledge  of  the  relation  between  an  equation 
and  the  graphical  representation  of  the  functional  correspondence 
determined  by  the  equation.  Such  a  knowledge  is  most  easily 
obtained  by  a  study  of  linear  equations  and  equations  of  the 
second  degree  together  with  their  corresponding  loci.  This  knowl- 
edge is  not  only  of  importance  to  the  student  of  applied  mathe- 
matics, but  it  has  a  special  disciplinary  value  for  the  general 
student. 

The  standard  forms  of  the  equations  of  a  number  of  important 
loci  are  developed  early  (Chapter  IV),  and  the  properties  of  these 
loci  are  discussed  in  detail  later  (Chapters  VI  and  VII)  by  means 
of  the  equations  already  at  hand.  By  this  arrangement,  it  is 
hoped  that  some  unnecessary  repetition  has  been  avoided. 

The  equations  of  tangents  to  the  conic  sections  have  been 
derived  by  means  of  the  discriminant  of  the  quadratic  equation 
whose  roots  are  the  ^'-coordinates  of  the  points  of  intersection 
with  a  variable  secant,  rather  than  by  means  of  the  derivative. 
This  course  has  been  adopted,  first,  because  the  geometric  inter- 


iv  PREFACE 

pretation  of  the  discriminant  is  important  in  itself ;  and,  second, 
because  the  use  of  the  derivative  ought,  logically,  to  be  preceded 
by  a  chapter  devoted  to  its  definition  and  the  methods  for  finding 
it,  at  least  for  algebraic  functions.  Moreover,  the  use  of  the 
derivative  for  finding  the  equations  of  tangents  is  only  one  of 
its  many  applications.  No  student  should  feel  that  his  mathe- 
matical education  is  complete  without  a  knowledge  of  the  calculus, 
where  he  will  become  familiar  with  the  derivative  and  can  appre- 
ciate its  usefulness  in  many  directions. 

The  present  volume  is  designed  for  a  four-hour,  or  a  five-hour, 
course  for  one  semester,  but  may  be  shortened  to  a  three-hour 
course  by  omitting  certain  parts  of  the  text.  For  example,  Art. 
105  may  be  omitted  without  marring  the  continuity  of  the  course. 
Again,  Arts.  110,  111,  and  112  contain  all  that  is  essential  in 
dealing  with  the  general  equation  of  the  second  degree  in  two 
variables,  and  the  remainder  of  Chapter  VIII  can  therefore  be 
omitted  from  the  longer  course.  Parts  of  Chapter  IX  can  also 
be  omitted  according  to  the  needs  of  the  student.  The  chapters 
on  solid  analytic  geometry  have  been  added  for  the  benefit  of 
those  students  who  have  time  only  for  an  outline  of  the  subject 
matter.    ISTo  apology  is  therefore  offered  for  the  meager  treatment. 

The  authors  desire  to  express  their  appreciation  to  their  col- 
leagues of  the  University  of  Wisconsin  and  of  the  University 
of  Illinois  for  the  assistance  and  the  many  helpful  suggestions 
given  them  during  the  preparation  of  the  book.  They  are  under 
especial  obligations  to  Professor  W.  H.  Bussey,  of  the  University 
of  Minnesota;  Professor  S.  C.  Davisson,  of  the  University  of 
Indiana ;  Professor  J.  L.  Markley,  of  the  University  of  Michigan ; 
and  Professor  E.  J.  Townsend,  of  the  University  of  Illinois,  for 
their  care  and  assistance  in  seeing  the  book  through  the  press. 

L.    W.    BOWLING, 
F.   E.    TURNEAURE. 
University  of  Wisconsin, 
July,  1914. 


CONTENTS 


INTRODUCTION 

PAGE 

A.  The  quadratic  equation 1 

B.  Trigonometric  formulas      .........         1 

C.  Numerical  tables 3 


PART 
PLANE  ANALYTIC   GEOMETRY 

CHAPTER  I 
SYSTEMS   OF   COORDINATES 

ARTICLE 

1.  The  linear  scale 7 

2.  Directed  segments,  directed  angles 8 

3.  Addition  of  directed  segments,  addition  of  directed  angles       .         .  9 

4.  Position  of  a  point  in  a  plane 10 

5.  Cartesian  coordinates        .........  10 

6.  Rectangular  coordinates 11 

7.  Notation 12 

8.  Polar  coordinates      ..........  13 

9.  Relation  between  rectangular  coordinates  and  polar  coordinates      .  14 


CHAPTER   II 

DIRECTED  SEGMENTS  AND  AREAS  OF  PLANE  FIGURES 

10.  Projections  upon  the  coordinate  axes 18 

11.  Inclination  and  slope  of  a  directed  segment 18 

12.  The  length  of  a  segment 20 

13.  Angle  which  one  segment  makes  with  another          ....  22 

14.  Parallel  segments 23 

15.  Perpendicular  segments ,23 

16.  Point  bisecting  a  given  segment 24 

17.  Point  dividing  a  given  segment  in  a  given  ratio        .         .         .         .25 

18.  Area  of  a  triangle,  one  vertex  at  the  origin 26 

19.  Sign  of  the  expression  (xi2/2  —  A'22/i) 26 

20.  Area  of  a  triangle,  vertices  in  any  position       .....  28 

21.  Area  of  any  polygon         .........  30 

V 


vi  CONTENTS 

CHAPTEE   III 
FUNCTIONS  AND   THEIR   GRAPHIC   REPRESENTATION 

ARTICLE  PAGE 

22.  Constants  and  variables 33 

23.  Functions 33 

24.  Notation 33 

25.  Determination  of  functional  correspondence    .....  33 

26.  Dependent  and  independent  variables 34 

27.  Graphic  representation      .........  34 

28.  Single-valued  and  multiple-valued  functions     .....  36 

29.  Symmetry 37 

30.  Intercepts 38 

31.  Graph  in  polar  coordinates        ........  39 

32.  Algebraic  functions 41 

33.  Transcendental  functions 41 

34.  Graphs  of  transcendental  functions  .......  41 

35.  Geometric  construction  of  the  graphs  of  trigonometric  functions      .  42 

36.  The  exponential  function 44 

37.  Graph  of  the  exponential  function 44 

38.  Inverse  functions c         .  45 

39.  Graph  of  an  inverse  function 46 

40.  Observation 48 

41.  Machines 48 

CHAPTER   IV 

LOCI   AND   THEIR   EQUATIONS 

42.  Locus  of  a  point,  equation  of  locus  .......  53 

■  43.    A  fundamental  problem 53 

44.  General  definition 54 

45.  The  circle 55 

46.  The  equation  :>i^  -\- if  +  Ax  +  By  -\-  C  =  0 56 

47.  The  straight  line       ..........  57 

48.  The  determinant  form 59 

49.  The  ellipse 60 

50.  The  axes  and  eccentricity 62 

51.  The  hyperbola .63 

52.  Axes  and  eccentricity       .........  65 

53.  The  parabola 66 

54.  The  cassinian  ovals .         .67 

55.  Recapitulation 69 

56.  Polar  equation  of  a  circle  .........  69 

57.  Polar  equation  of  a  straight  line        ....                  .         .  70 


CONTENTS 


Vll 


ARTICLE 

58.  Polar  equation  of  the  parabola  ..... 

59.  Polar  equations  of  the  ellii^se  and  the  hyperbola 

60.  Parametric  equations 

61.  Geometrical  construction  of  the  ellipse  and  the  hyperbola 

62.  Recapitulation 


71 
71 
73 
74 

76 


CHAPTER   V 

EQUATIONS   AND   THEIR  LOQI 

63.  Locus  of  an  equation         .... 

64.  A  second  fundamental  problem 

65.  Discussion  of  an  equation 

66.  Example  I.     The  equation  x-  +  4  y^  =  4  . 

67.  Example  II,     Tlie  equation  x'^  —  iy-  =4: 

68.  Example  III.     The  equation  xy  —  x  —  y  =  0 

69.  Example  IV.     The  equation  y  — 

70.  Example  V.     The  equation  y^  = 


1  +x- 
x'(6— x) 
3x  -\-  b 


Example  VI. 


The  catenary,  y  =  ^(e" 


) 


Simple      harmonic      curves,      compound 


72.  Example     VII. 

curves         ...... 

73.  Example  VIII.     Damped  vibrations 

74.  Polar  equations         .         .         . 

75.  Example  IX.     The  equation  r  =  cos  2  e  . 

76.  Example  X.     The  equation  r-  =  a^  cos  2  6 


77 
77 
77 
79 
80 
81 

82 
83 


harmonic 
.       86 

.       87 


89 
90 


TRANSFORMATION  OF  COORDINATES 

77.  Translation  of  the  axes 91 

78.  Rotation  of  the  axes  .........  92 

79.  Removal  of  terms  of  first  degree 94 

80.  Removal  of  the  term  in  xy 95 

81.  Classification  of  algebraic  curves      .         • 96 


CHAPTER   VI 
LOCI   OF  FIRST   ORDER 


82.  Linear  equations 

83.  Intersection  of  two  lines  . 

84.  The  pencil  of  lines    . 

85.  The  pair  of  hues 


98 

99 

100 

102 


viii  CONTENTS 

ARTICLE  PAGE 

86.  The  normal  form 102 

87.  Reduction  of  Ax  +  By  +  C  —  0  to  the  normal  form        .         .         .  103 

88.  Distance  from  a  line  to  a  point 104 

89.  The  angle  which  one  line  makes  with  another         ....  105 


CHAPTER   VII 

LOCI  OF  SECOND  ORDER.     EQUATIONS  IN  STANDARD 
FORM 

DIRECTRICES 

90.  Review 107 

91.  Directrices 108 

92.  A  fundamental  theorem 108 

93.  Construction  of  an  ellipse  or  an  hyijerbola     .....  109 

94.  Two  common  properties 110 

TANGENTS 

95.  Equation  of  a  tangent  in  terms  of  the  slope  .         .         .         .         .111 

96.  Coordinates  of  the  point  of  contact 11-5 

97.  Equation  of  a  tangent  in  terms  of  the  coordinates  of  the  point  of 

contact     .         .         .        .        .         .         .         ,         .         .         .116 

98.  Normals 110 

99.  Tangent  length,  normal  length,  sub-tangent,  sub-normal       .         .  119 

100.  Reflection  properties 120 

DIAMETERS 

101.  Definition  of  diameter     ..........  123 

102.  Conjugate  diameters 123 

103.  The  locus  of  the  middle  points  of  a  system  of  parallel  chords         .  12-5 

POLES   AND   POLAR  LINES 

104.  Definition  of  pole  and  polar  line      .......  127 

105.  Geometric  properties  of  poles  and  polar  lines          ....  129 

SYSTEMS  OF  CONICS 

106.  The  asymptotes  of  the  hyperbola    .......  133 

107.  Conjugate  hyperbolas      ..•....,.  134 

108.  The  system  of  concentric  hyperbolas       ......  135 

109.  The  system  of  confocal  conies         .......  136 


CONTENTS 


IX 


0 


CHAPTER   VIII 

LOCI  OF  THE  SECOND    ORDER.     EQUATIONS   NOT  IN 
STANDARD  FORM 

ARTICLE 

110.  Translation  of  the  coordinate  axes 

111.  Discussion  of  the  equation  ax'^  +  hy^  +  2  gx  -\-  2fy  +  c 

112.  The  general  equation  of  the  second  degree 

113.  Rsmoval  of  the  terms  of  first  degi'ee 

114.  First  case,  «?) —/i- ^t  0    .... 

115.  Second  case,  ah  —  li^  =  0,  hf  —  hg  ^  0  . 

116.  Third  case,  ah  -  h'^  =  0  and  hf  -  bg  =  0 

117.  Recapitulation 


PAGE 

141 
142 
145 
148 
149 
152 
155 
156 


TANGENTS  AND  DIAMETERS 


118.  Tangents 

119.  Diameters 


157 
159 


SYSTEMS  OF  CONICS 


The  pencil  of  conies        ..... 
The  system  of  circles  with  a  common  radical  axis 
The  parabolas  in  the  pencil  U+  k\'  =  0 
Straight  lines  in  the  pencil  U  +  Vk  —  0 


16.3 
163 
164 

165 
167 


CHAPTER   IX 


LOCI   OF  HIGHER   ORDER  AND   OTHER  LOCI 
125.    Introductory  note 


169 


126. 
127. 
128. 
129. 


130. 
131. 
132. 
133. 
134 


ALGEBRAIC   LOCI 

The  Cissoid  of  Diodes    .... 
The  Conchoid  of  Nicomedes    . 
The  Witch  of  Agnesi       .... 
The  Liraagon  of  Pascal  .... 


TRANSCENDENTAL  LOCI 


The  cycloid 
The  hypocycloid 
Special  hypocycloids 
The  epicycloid 
The  cardioid    . 


169 
170 
171 
172 


173 
175 
176 
177 
178 


X  CONTENTS 

EMPIRICAL  EQUATIONS  AND  THEIR  LOCI 

ARTICLE  PAGE 

135.  Typical  equations 182 

136.  Loci  of  typical  equations 182 

137.  Selection  of  type-curve  and  determination  of  constants  .         .  186 

138.  Test  by  means  of  linear  equations 187 

139.  Examples  and  exercises 189 

140.  'Type,  y  =  a +  bx  +  cx^  +  dx^  +  ■■■  ^kx" 193 

PART  n 
SOLID   ANALYTIC   GEOMETRY 


CHAPTER   X 
SYSTEMS   OF   COORDINATES 


141.  Rectangular  and  oblique  coordinates 

142.  Spherical  coordinates 

143.  Cylindrical  coordinates  . 


195 
196 
197 


CHAPTER   XI 

DIRECTED   SEGMENTS  IN  SPACE 

144.  Projections  upon  the  coordinate  axes 

145.  Length  of  a  segment 

146.  Direction  angles  and  direction  cosines  of  a  segment 

147.  Relation  connecting  the  direction  cosines  of  a  segment 

148.  Projection  of  a  segment  upon  any  line    . 

149.  Projection  of  a  broken  line 

150.  The  angle  between  two  segments    . 

151.  Perpendicular  segments,  parallel  segments 

152.  Point  dividing  a  given  segment  in  a  given  ratio 


199 
199 
200 
200 
202 
202 
203 
203 
204 


CHAPTER   XII 

LOCI  AND   THEIR  EQUATIONS 

153.  Surfaces  and  curves 

154.  Equations  of  loci     . 

155.  The  sphere 

156.  Surfaces  of  revolution 

157.  Cylinders 

158.  The  right  circular  cone 

159.  Plane  sections  of  a  right  circular  cone 


207 
207 
208 
209 
210 
211 
211 


CONTENTS 


XI 


CHAPTER   XIII 
THE  PLANE  AND  THE  STRAIGHT  LINE  IN   SPACE 

ABTICLE  PAGE 

160.  The  normal  form  of  the  equation  of  a  plane 213 

161.  The  intercept  form  of  the  equation 215 

162.  Equation  of  a  plane  through  three  given  points     ....  215 

163.  Determinant  form  of  the  equation 216 

164.  Perpendicular  distance  from  a  plane  to  a  point       ....  217 

165.  Angle  between  two  planes       ........  218 

166.  Pencil  of  planes  with  a  common  axis 219 

167.  Pencil  of  planes  with  a  common  vertex 220 

168.  The  equations  of  a  straight  line  in  space 221 

169.  The  projecting  planes  of  a  line 222 

170.  The  intersection  of  two  planes 223 

171.  Intersection  of  a  line  with  a  plane 225 


CHAPTER   XIV 


EQUATIONS   AND   THEIR   LOCI 

172.  Second  fundamental  problem 

173.  Construction  of  a  surface  from  its  equation 

174.  The  quadric  surfaces,  or  conicoids 

175.  The  ellipsoid  ... 

176.  The  hyperboloid  of  one  sheet . 

177.  The  hyperboloid  of  two  sheets 

178.  The  elliptic  paraboloid     . 

179.  The  hyperbolic  paraboloid 

180.  The  quadric  cone    . 

181.  CyUnders 

182.  Pairs  of  planes 

183.  Ruled  surfaces 

184.  Equation  of  generator     . 

185.  Tangent  lines  and  planes 

186.  Circular  sections 

187.  Asymptotic  cones     . 

188.  Projecting  cylinders  of  a  curve  in  space 

189.  Parametric  equations  of  curves  in  space 

190.  The  circular  helix   ,         ,         .         .         . 

Answers    .    .    ,    .    .    .    . 


228 
228 
229 
230 
231 
233 
234 
235 
236 
237 
237 
238 
239 
240 
242 
243 
244 
245 
246 

249 


GREEK   ALPHABET 


Letteks 

Letteks 

Names 

Names 

Capitals 

Lower 

Case 

Capitals 

Lower 

Case 

A 

a 

Alpha 

N 

V 

Nu 

B 

P 

Beta 

E 

$ 

Xi 

r 

7 

Gamma 

0 

o 

Omicron 

A 

8 

Delta 

n 

TT 

Pi 

E 

e 

Epsilon 

p 

P 

Rho 

Z 

^ 

Zeta 

2 

a 

Sigma 

H 

V 

Eta 

T 

T 

Tau 

© 

0 

Theta 

Y 

V 

Upsilon 

I 

L 

Iota 

$ 

4> 

Phi 

K 

K 

Kappa 

X 

X 

Chi 

A 

X 

Lambda 

* 

^ 

Psi 

M 

H- 

Mil 

o 

<x) 

Omega 

2U 


ANALYTIC   GEOMETRY 


INTRODUCTION 

A.    The  quadratic  equation.     For  coiwenience  in  reference  the 
following  formulas  and  tables  are  introduced. 

Any  quadratic  equation  may  be  written  in  the  form 

a.i-2  +  bx  +  c  =  0. 
The  two  roots  of  this  equation  are 


^i  = 7i ■■ — ,  and  ic2  == . 

By  addition,  x^  +  x.,  = . 

a 

By  multiplication,  x^x^  =  -. 

a 

The  sum  and  the  product  of  the  roots  can  therefore  be  found 
directly  from  the  equation  without  solving. 

The  character  of  the  roots  depends  on  the  quantity  under  the 
radical,  ft^  _  4  «(.. 

If  6-  —  4 ac> 0,  the  roots  are  real  and  unequal, 
if  6^  —  4  ac  =  0,  the  roots  are  real  and  equal, 
if  6^  —  4  ac  <  0,  the  roots  are  imaginary. 

The  expression  b-  —  4  ac  is  called  the  discriminant  of  the 
equation,  and  when  placed  equal  to  zero  expresses  the  condition 
which  must  hold  between  the  coefficients,  if  the  two  roots  of  the 
equation  are  equal. 

B.  Trigonometric  formulas.  If  A,  B,  and  C  are  the  angles  of  a 
triangle  and  a,  b,  c  are  respectively  the  lengths  of  the  sides 
opposite,  then  : 

1 


INTRODUCTION 

(1)  Laiv  of  sines: 

sin  A  _  siii^  _  sin  C 
a  b  c    ' 

(2)  Laiv  of  cosines  : 

a'^=  b^  +  c^  —  2  be  cos  A, 

b^  =  a^  +  c^  —  2ac  cos  B, 
c'^  =  a'^  +  b'^  —  2  ab  cos  C. 

(3)  Lnw  of  tangents : 

tan \{A-B)      «t  -  6' 

Addition  formulas.     If  A  and  B  are  any  angles,  then 

sin  {A±B)  =  sin  ^  cos  J5  i  sin  B  cos  ^, 
cos  (^  ±  jE»)  =  cos  ^  cos  J5  =F  sin  A  sin  B, 
tan  4  ±  tan  B 


tan  (^  ±  B) : 


1  T  tan  ^  tan  B 


TABLES 


C.    TABLES 

Common  Logarithms 


N 
10 

o 

0000 

D 
43 

1 

0043 

D 

43 

2 

0086 

D 

42 

3 

D 

42 

4 

0170 

D  5 

D 

41 

6 

0253 

D 

41 

7 

D 

40 

8 

0334 

D 

40 

9 

0374 

D 

40 

0128 

42  0212 

0294 

11 

0414 

39 

0453 

39 

0492 

39 

0531 

38 

0569 

38  0607 

38 

0645 

37 

0682 

37 

0719 

36 

0755 

37 

12 

0792 

36 

0828 

36 

0864 

35 

0899 

35 

0934 

35  0969 

35 

1004 

34 

1038 

34 

1072 

34 

1106 

33 

13 

1139 

34 

1173 

33 

1206 

33 

1239 

32 

1271 

32  1303 

32 

1335 

32 

1367 

32 

1399 

31 

1430 

31 

14 
15 

1461 
1761 

31 
29 

1492 
1790 

31 

28 

1523 

30 
29 

1553 

1847 

31 

28 

1584 

1875 

30  1614 
28  1903 

30 

28 

1644 
1931 

29 
28 

1673 
1959 

30 
28 

1703 

1987 

29 

27 

1732 

29 
27 

1818 

2014 

16 

2041 

27 

2068 

27 

2095 

27 

2122 

26 

2148 

27  2175 

26 

2201 

26 

2227 

26 

2253 

26 

2279 

25 

17 

2304 

26 

2330 

25 

2355 

25 

2380 

25 

2405 

25  2430 

25 

2455 

25 

2480 

24 

2504 

25 

2529 

24 

18 

2553 

24 

2577 

24 

2601 

24 

2625 

23 

2648 

24  2672 

23 

2695 

23 

2718 

24 

2742 

23 

2765 

23 

19 
20 

2788 
3010 

22 

2810 
3032 

23 
22 

2833 
3054 

23 
21 

2856 
3075 

22 
21 

2878 
3096 

22  2900 

22  3118 

23 

21 

2923 
3139 

22 
21 

2945 
3160 

21 

2967 
3181 

22 
20 

2989 

21 
21 

3201 

21 

3222 

21 

3243 

20 

3263 

21 

3284J20 

3304 

20  3324 

21 

3345 

20 

3365 

20 

3385 

19 

3404 

20 

22 

3424 

20 

3444 

20 

3464 

19 

3483119 

3502 

>o  3522 

19 

3541 

19 

3560 

19 

3579 

19 

3598 

19 

23 

3617 

19 

3636 

19 

3655 

19 

3674!  IS 

3692 

L9  3711 

18 

3729 

18 

3747 

19 

3766 

18 

3784 

18 

24 
25 

3802 

18 
18 

3820 

18 
17 

3838 
4014 

18 
17 

3856  18 

3874 
4048 

8  3892 
L7  4065 

17 
17 

3909 
4082 

18 
17 

3927 
4099 

18 
17 

3945 
4116 

17 

17 

3962 
4133 

17 

17 

3979 

3997 

4031 

17 

26 

4150 

16 

4166 

17 

4183 

17 

4200 

16 

4216 

16  4232 

17 

4249 

16 

4265 

16 

4281 

17 

4198 

16 

27 

4314 

16 

4330 

16 

4346 

16 

4362 

16 

4378 

5  4393 

16 

4409 

16 

4425 

15 

4440 

16 

4456 

16 

28 

4472 

15 

4487 

15 

4502 

16 

4518 

15 

4533 

L5  4548 

16 

4564 

15 

4579 

15 

4594 

15 

4609 

15 

29 
30 

4624 
4771 

15 

15 

4639 

4786 

15 
14 

4654 
4800 

15 
14 

4669 
4814 

14 
15 

4683 

5  4698 
4  4843 

15 
14 

4713 
4857 

15 
14 

4728 
4871 

14 

4742 
4886 

15 
U 

4757 

14 
14 

4829 

4900 

31 

4914 

14 

4928 

14 

4942 

13 

4955 

14 

4969 

4  4983 

14 

4997 

14 

5011 

13 

5024 

14 

5038 

13 

32 

5051 

14 

5065 

14 

5079 

13 

5092 

13 

5105 

4  5119 

13 

5132 

13 

5145 

14 

5159 

13 

5172 

13 

33 

5185 

13 

5198 

13 

5211 

13 

5224 

13 

5237 

3  5250 

13 

5263 

13 

5276 

13 

5289 

13 

5302 

13 

34 
35 

5315 
5441 

13 
12 

5328 
5453 

12 
12 

5340 
5465 

13 
13 

5353 
5478 

13 

12 

5366 
5490 

2  5378 
2  5502 

13 

12 

5391 
5514 

12 
13 

5403 

5527 

13 
12 

5416 

12 
12 

5428 

13 
12 

5539 

5551 

36 

5563 

12 

5575 

12 

5587 

12 

5599 

12 

5611 

2  5623 

12 

5635 

12 

5647 

11 

5658 

12 

5670 

12 

37 

5682 

12 

5694 

11 

5705 

12 

5717 

12 

5729 

1  5740 

12 

5752 

11 

5763 

12 

5775 

11 

5786 

12 

38 

5798 

11 

5809 

12 

5821 

11 

5832 

11 

5843 

2  5855 

11 

5866 

11 

5877 

11 

5888 

11 

5899 

12 

39 
40 

5911 
6021 

11 
10 

5922 

11 
11 

5933 
6042 

11 
11 

5944 
6053 

11 
11 

5955  ] 
6064  1 

1  5966 

1] 
10 

5977 
6085 

11 
11 

5988 
6096 

11 
11 

5999 
6107 

11 
10 

6010 
6117 

11 
11 

6031 

16075 

41 

6128 

10 

6138 

11 

6149 

11 

6160 

10 

6170] 

0  6180 

11 

6191 

10 

6201 

11 

6212 

10 

6222 

10 

42 

6232 

11 

6243 

10 

6253 

10 

6263 

11 

6274  1 

0  6284 

10 

6294 

10 

6304 

10 

6314 

11 

6325 

10 

43 

6335 

10 

6345 

10 

6355 

10 

6365 

10 

6375 

0  6385 

10 

6395 

10 

6405 

10 

6415 

10 

6425 

10 

44 
45 

6435 
6532 

9 
10 

6444 

10 
9 

6454 
6551 

10 
10 

6464 
6561 

10 
10 

6474 
6571 

0  6484 
9  6580 

9 
10 

6493 
6590 

10 
9 

6503 
6599 

10 
10 

6513 
6609 

9 
9 

6522 

10 
10 

6542 

6618 

46 

6628 

9 

6637 

9 

6646 

10 

6656 

9 

6665  ] 

0  6675 

9 

6684 

9 

6693 

9 

6702 

10 

6712 

9 

47 

6721 

9 

6730 

9 

6739 

10 

6749 

9 

6758 

9  6767 

9 

6776 

9 

6785 

9 

6794 

9 

6803 

9 

48 

6812 

9 

6821 

9 

6830 

9 

6839 

9 

6848 

9  6857 

9 

6866 

9 

6875 

9 

6884 

9 

6893 

9 

49 
50 

6902 
6990 

9 
8 

6911 

6998 

9 
9 

6920 
7007 

8 
9 

6928 
7016 

9 
8 

6937 
7024 

9  6946 
9  7033 

9 
9 

6955 
7042 

9 
8 

6964 
7050 

8 
9 

6972 
7059 

9 

8 

6981 

9 
9 

7067 

51 

7076 

8 

7084 

9 

7093 

8 

7101 

9 

7110 

8  7118 

8 

7126 

9 

7135 

8 

7143 

9 

7152 

8 

52 

7160 

8 

7168 

9 

7177 

8 

7185 

8 

7193 

9  7202 

8 

7210 

8 

7218 

8 

7226 

9 

7235 

8 

53 

7243 

8 

7251 

8 

7259 

8 

7267 

8 

7275 

9  7284 

8 

7292 

8 

7300 

8 

7308 

8 

7316 

8 

54 

7324 

8 

7332 

8 

7340 

8 

7348 

8 

7356 

8  7364 

8 

7372 

8 

7380 

8 

7388 

8 

7396 

8 

TABLES 


Common  Logarithms  —  Continued 


N 

55 

56 
57 
58 
59 

60 

61 
62 
63 
64 

65 

66 
67 
68 
69 

70 

71 
72 
73 

74 

75 

76 
77 
78 
79 

80 

81 
82 
83 
84 

85 

86 
87 
88 
89 

90 

91 
92 
93 
94 

95 

96 
97 
98 
99 

o 

7404 
7482 
7559 
7634 
7709 

7782 
7853 
7924 
7993 
8062 

8129 
8195 
8261 
8325 
8388 

8451 
8513 

8573 
8633 
8692 

8751 
8808 
8865 
8921 
8976 

9031 
9085 
9138 
9191 
9243 

9294 
9345 
9395 
9445 
9494 

9542 
9590 
9638 
9685 
9731 

9777 
9823 
9868 
9912 
9956 

u 

8 
8 
7 
8 
7 

7 
7 
7 
7 
7 

7 
7 
6 
6 

7 

6 
6 
6 
6 
6 

5 
6 
6 
6 
6 

5 
5 
5 
5 
5 

5 
5 
5 
5 

5 

5 

5 

5 
4 
4 
5 
5 

1 

7412 
7490 
7566 
7642 
7716 

7789 
7860 
7931 
8000 
8069 

8136 
8202 
8267 
8331 
8395 

8457 
8519 
8579 
8639 
8698 

8756 
8814 
8871 
8927 
8982 

9036 
9090 
9143 
9196 
9248 

9299 
9350 
9400 
9450 
9499 

9547 
9595 
9643 
9689 
9736 

9782 
9827 
9872 
9917 
9961 

D 

7 
7 
8 
7 
7 

7 
8 
7 
7 
6 

6 
7 
7 
7 
6 

6 

6 
6 
6 
6 

6 
6 
5 
5 
5 

6 
6 
6 
5 
5 

5 
5 
5 
5 
5 

5 
5 
4 
5 
5 

4 
5 
5 

4 
4 

2 

7419 
7497 
7574 
7649 
7723 

7796 

7868 
7938 
8007 
8075 

8142 
8209 
8274 
8338 
8401 

8463 

8525 
8585 
8645 
8704 

8762 
8820 
8876 
8932 
8987 

9042 
9096 
9149 
9201 
9253 

9304 
9355 
9405 
9455 
9504 

9552 
9600 
9647 
9694 
9741 

9786 
9832 
9877 
9921 
9965 

D 

8 
8 
8 

7 

7 

7 
6 
6 
6 
6 

7 
6 
6 
6 
6 

6 
5 
6 
6 
6 

5 
5 
5 

5 
5 

5 
6 
5 
5 
6 

5 

5 

5 

4 

^ 

4 

4 
5 

3 

7427 
7505 
7582 
7657 
7731 

7803 
7875 
7945 
8014 

8082 

8149 
8215 
8280 
8344 

8407 

8470 
8531 
8591 
8651 
8710 

8768 
8825 
8882 
8938 
8893 

9047 
9101 
9154 
9206 
9258 

9309 
9360 
9410 
9460 
9509 

9557 
9605 
9652 
9699 
9745 

9791 
9836 
9881 
9926 
9969 

D 

8 
8 
7 
7 
7 

7 
7 
7 
7 
7 

' 

7 

7 

7 
7 

6 
6 
6 
6 
6 

6 
6 

5 
5 
5 

6 

5 
5 
6 
5 

6 
5 
5 
5 

4 

5 
4 
5 
4 
5 

4 
5 
5 
4 
5 

4 

7435 
7513 
7589 
7664 

7738 

7810 
7882 
7952 
8021 
8089 

8156 
8222 
8287 
8351 
8414 

8476 
8537 
8597 
8657 
8716 

8774 
8831 
8887 
8943 
8998 

9053 
9106 
9159 
9212 
9263 

9315 
9365 
9415 
9465 
9513 

9562 
9609 
9657 
9703 
9750 

9795 
9841 
9886 
9930 
9974 

D 

8 
7 
8 
8 
7 

8 

7 
7 

6 
6 
6 
6 
6 

6 
6 
6 
6 
6 

5 
6 
6 
6 
6 

5 
6 
6 
5 
6 

5 
5 
5 
4 
5 

4 
5 
4 
5 
4 

5 

4 
4 
4 
4 

5 

7443 

7520 
7597 

7672 
7745 

7818 
7889 
7959 
8028 
8096 

8162 

8228 
8293 
8357 
8420 

8482 
8543 
8603 
8663 

8722 

8779 
8837 
8893 
8949 
9004 

9058 
9112 
9165 
9217 
9269 

9320 
9370 
9420 
9469 

9518 

9566 
9614 
9661 
9708 
9754 

9800 
9845 
9890 
9934 
9978 

D 

8 
8 
7 
7 
7 

7 

7 
7 
7 
6 

7 
7 
6 
6 
6 

6 
6 
6 
6 
5 

6 
5 
6 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 

4 
5 
5 

6 

7451 

7528 
7604 
7679 

7752 

7825 
7896 
7966 
8035 
8102 

8169 
8235 
8299 
8363 
8426 

8488 
8549 
8609 
8669 

8727 

8785 
8842 
8899 
8954 
9009 

9063 
9117 
9170 
9222 
9274 

9325 
9375 
9425 
9474 
9523 

9571 
9619 
9666 
9713 
9759 

9805 
9850 
9894 
9939 
9983 

D 

8 
8 
8 
7 
8 

7 
7 
7 
6 

7 

7 
6 
7 
7 
6 

6 
6 
6 
6 
6 

6 
6 
5 
6 
6 

6 
5 
5 
5 
5 

5 
5 
5 
5 
5 

5 
5 

5 
4 
4 

4 
4 
5 

4 
4 

7 

7459 
7536 
7612 
7686 
7760 

7832 
7903 
7973 
8041 
8109 

8176 
8241 
8306 
8370 
8432 

8494 
8555 
8615 

8675 
8733 

8791 
8848 
8904 
8960 
9015 

9069 
9122 
9175 
9227 
9279 

9330 
9380 
9430 
9479 

9528 

9576 
9624 
9671 
9717 
9763 

9809 
9854 
9899 
9943 
9987 

D 

7 
7 
7 
8 
7 

7 
7 
7 
7 
7 

6 

7 
6 
6 
7 

6 
6 
6 
6 
6 

6 
6 
6 
5 
5 

5 
6 
5 
5 
5 

5 
5 
5 
5 

5 

5 
4 
4 
5 
5 

5 
5 

4 
5 

8 

7466 
7543 
7619 
7694 

7767 

7839 
7910 
7980 
8048 
8116 

8182 
8248 
8312 
8376 
8439 

8500 
8561 
8621 
8681 
8V39 

8797 
8854 
8910 
8965 
9020 

9074 
9128 
9180 
9232 
9284 

9335 
9385 
9435 
9484 
9533 

9581 
9628 
9675 
9722 
9768 

9814 
9859 
9903 
9948 
9991 

D 

8 
8 
8 

7 
7 

7 
7 
7 
7 
6 

7 
6 
7 
6 
6 

6 
6 
6 
5 
6 

5 
5 
5 
6 
5 

5 
5 
6 
6 
5 

5 
5 
5 
5 
5 

5 
5 
5 
5 
5 

4 
4 
5 
4 
5 

9 

7474 
7551 
7627 
7701 
7774 

7846 
7917 
7987 
8055 
8122 

8189 
8254 
8319 
8382 
8445 

8506 

8567 
8627 
868(5 
8745 

8802 
8859 
8915 
8971 
9025 

9079 
9133 
9186 
9238 
9289 

9340 
9390 
9440 
9489 
9538 

9586 
9633 
9680 
9727 
9773 

9818 
9863 
9908 
9952 
9996 

D 

8 
8 
7 
8 
8 

7 
7 
6 
7 
7 

6 
7 
6 
6 
6 

7 
6 
6 
6 
6 

6 
6 
6 
5 
6 

6 
5 
5 
5 

5 

5 
5 
5 
5 

4 

4 
5 
5 
4 
4 

5 
5 

4 
4 
4 

TABLES 


Trigonometric  Functions 

[Characteristics  of  Logarithms  omitted  —  determine  by  the  usual  rule  from  the  value] 


Radians 

De- 

Sine 

Tangent 

Cotangent 

Cosine 

grees 

Talue 

logio 

Value  login 

Value  logio 

Value  logio 

.0000 

0° 

.0000 

—  CO 

.0000  -00 

00       00 

1.0000  GOOO 

90° 

1..5708 

.0175 

1° 

.0175 

2419 

.0175  2419 

57.290  7581 

.9998  9999 

89° 

1.5533 

.0349 

2° 

.0349 

5428 

.0.349  .5431 

28.036  4569 

.9994  9997 

88^ 

1.5359 

.0524 

3° 

.0523 

7188 

.0524  7196 

19.081  2806 

.9986  9994 

87° 

1.5184 

.0698 

4° 

.0698 

8436 

.0699  8448 

14.301  1554 

.9976  9989 

86° 

1.5010 

.0873 

5° 

.0872 

9403 

.0875  9420 

11.4.30  0580 

.9962  9983 

85° 

1.48.35 

.1047 

6° 

.1045 

0192 

.1051  0216 

9.5144  9784 

.9945  9976 

84° 

1.4661 

.1222 

7° 

.1219 

0859 

.1228  0891 

8.1443  9109 

.9925  9968 

83° 

1.4486 

.1396 

8° 

.1392 

1436 

.1405  1478 

7.1154  8522 

.9903  9958 

82° 

1.4312 

.1571 

9° 

.1564 

1943 

.1584  1997 

6.3138  8003 

.9877  9946 

81° 

1.4137 

.1745 

10° 

.1736 

2397 

.1763  2463 

5.6713  7537 

.9848  9934 

80° 

1.3963 

.1920 

11° 

.1908 

2806 

.1944  2887 

5.1446  7113 

.9816  9919 

79° 

1.3788 

.2094 

12° 

.2079 

3179 

.2126  3275 

4.7046  6725 

.9781  9904 

78° 

1.3614 

.2269 

1.3° 

.2250 

3521 

.2309  36.34 

4.3315  6366 

.9744  9887 

77° 

1.3439 

.2443 

14° 

.2419 

3837 

.2493  3968 

4.0108  6032 

.9703  98(59 

76° 

1.3265 

.2618 

1.5° 

.2.588 

4130 

.2679  4281 

3.7321  5719 

.9659  9849 

75° 

1 .3090 

.2793 

16° 

.2756 

4403 

.2867  4575 

3.4874  5425 

.9613  9828 

74'- 

1.2915 

.2967 

17° 

.2924 

4659 

..3057  4853 

3.2709  5147 

.9563  9806 

73° 

1.2741 

.3142 

18° 

.3090  4900 

.3249  5118 

3.0777  4882 

.9511  9782 

72° 

1.2566 

.3316 

19° 

.3256 

5126 

.3443  5370 

2.9042  4030 

.9455  9757 

71° 

1.2392 

.3491 

20° 

.3420 

5341 

.3640  5611 

2.7475  4389 

.9397  9730 

70° 

1.2217 

.3665 

21° 

.3584 

5543 

.38.39  2842 

2.0051  4158 

.9336  9702 

69° 

1.2043 

.3840 

22° 

.3746 

5736 

.4040  6064 

2.4751  .3936 

.9272  9672 

68° 

1.1868 

.4014 

23° 

.3907 

5919 

.4245  6279 

2.3559  3721 

.9205  9640 

67° 

1.1694 

.4189 

24° 

.4067 

6093 

.4452  6486 

2.2460  3514 

.9135  9607 

66° 

1.1519 

.4363 

25° 

.4226 

6259 

.4663  6687 

2.1445  3313 

.9063  9573 

65° 

1.1.345 

.4538 

26° 

.4384 

6418 

.4877  6882 

2.0503  3118 

.8988  9537 

64° 

1.1170 

.4712 

27° 

.4540 

6570 

.5095  7072 

1.9626  2928 

.8910  9499 

63° 

1.0996 

.4887 

28° 

.4695 

6716 

.5317  7257 

1.8807  2743 

.8829  9459 

62° 

1.0821 

.6061 

29° 

.4848 

6856 

.5543  7438 

1.8040  2562 

.8746  9418 

61° 

1.0647 

.5236 

30° 

.5000 

6990 

.5774  7614 

1.7321  2.386 

.8660  9375 

60° 

1.0472 

.5411 

31° 

.5150 

7118 

.6009  7788 

1.6643  2212 

.8572  9331 

59° 

1.0297 

.5585 

.32° 

.5299 

7242 

.6249  79.58 

1.6003  2042 

.8480  9284 

58° 

1.0123 

.5760 

33° 

.5446 

7361 

.6494  8125 

1.5399  1875 

.8387  92.36 

57° 

.9948 

.5934 

34° 

.5592 

7476 

.6745  8290 

1.4826  1710 

.8290  9186 

56° 

.9774 

.6109 

35° 

.5736 

7586 

.7002  8452 

1.4281  1548 

.8192  9134 

55° 

.9599 

.6283 

36° 

.5878 

7692 

.7265  8613 

1.3764  1387 

.8090  9080 

54° 

.9425 

.6458 

37° 

.6018 

7795 

.7536  8771 

1.3270  1229 

.7986  9023 

53° 

.9250 

.66.32 

38° 

.6157 

7893 

.7813  8928 

1.2799  1072 

.7880  8965 

52° 

.9076 

.6807 

39° 

.6293 

7989 

.8098  9084 

1.2349  0916 

.7771  8905 

51° 

.8901 

.6981 

40° 

.6428 

8081 

.8391  9238 

1  1918  0762 

.7660  8843 

50° 

.8727 

.7156 

41° 

.6561 

8169 

.8693  9392 

1.1504  0608 

.7547  8778 

49° 

.8552 

.7330 

42° 

.6691 

8255 

.9004  9544 

1.1106  0456 

.7431  8711 

48° 

.8378 

.7505 

43° 

.6820 

8338 

.9325  9697 

1.0724  0303 

.7314  8641 

47° 

.8203 

.7679 

44° 

.6947 

8418 

.96.57  9848 

1.0355  0152 

.7193  8569 

46° 

.8029 

.7854 

45° 

.7071 

8495 

1.0000  0000 

1.0000  0000 

.7071  8495 

45° 

.7854 

Value 

login 

Value  logio 

Value  logio 

Value  logjo 

De- 

Radians 

Cosine 

Cotangent 

Tangent 

Sine 

grees 

TABLES 

Exponential  Functions 


e^ 

e-^ 

e 

n 

e-^ 

X 

iogeX 

Value     logio 

Value     log-io 

X 

logeo; 

Value 

logio 

Value     logj^o 

0.0 

—  00 

1.000  0.000 

1.000  0.000 

2.0 

0.693 

7.389  0.869 

0.135  9.131 

0.1 

-2.303 

1.105  0.043 

0.905  9.957 

2.1 

0.742 

8.166  0.912 

0.122  9.088 

0.2 

-1.610 

1.221  0.087 

0.819  9.913 

2.2 

0.788 

9.025  0.955 

0.111  9.045 

0.3 

-1.204 

1.350  0.130 

0.741  9.870 

2.3 

0.833 

9.974  0.999 

0.100  9.001 

0.4 

-0.916  j  1.492  0.174 

0.670  9.826 

2.4 

0.875 

11.02 

1.023 

0.091  8.958 

0.5 

-0.693' 1.649  0.217 

0.607  9.783 

2.5 

0.916 

12.18 

1.086 

0.082  8.914 

0.6 

-0.511  11.822  0.261 

0.549  9.739 

2.6 

0.966 

13.46 

1.129 

0.074  8.871 

0.7 

-0.357   2.014  0.304 

0.497  9.696 

2.7 

0.993 

14.88 

1.173 

0.067  8.827 

0.8 

-0.223  12.226  0.347 

0.449  9.653 

2.8 

1.030 

16.44 

1.216 

0.061  8.784 

0.9 

-0.105 

2.460  0.391 

0.407  9.609 

2.9 

1.065 

18.17 

1.259 

0.055  8.741 

1.0 

0.000 

2.718  0.434 

0.368  9.566 

3.0 

1.099 

20.09 

1.303 

0.050  8.697 

1.1 

0.095  i  3.004  0.478 

0.333  9.522 

3.5 

1.253 

33.12 

1.520 

0.030  8.480 

1.2 

0.182 

3.320  0.521 

0.301  9.479 

4.0 

1.386 

54.60 

1.737 

0.018  8.263 

1.3 

0.262 

3.669  0.565 

0.273  9.435 

4.5 

1.504 

90.02 

1.954 

0.011  8.046 

1.4 

0.336 

4.055  0-608 

0.247  9.392 

5.0 

1.609 

148.4 

2.171 

0.007  7.829 

1,5 

0.405 

4.482  0.651 

0.223  9.349 

6.0 

1.792 

403.4 

2.606 

0.002  7.394 

1.6 

0.470 

4.953  0.695 

0.202  9.305 

7.0 

1.946 

1096.6  3.040 

0.001  6.960 

1.7 

0.531 

5.474  0.738 

0.183  9.262 

8.0 

2.079 

2981.0  3.474 

0.000  6.526 

1.8 

0.588 

6.050  0.782 

0.165  9.218 

9.0 

2.197 

8103.1  3.909 

0.000  6.091 

1.9 

0.642 

6.686  0.825 

0.150  9.175 

10.0 

2.303 

22026. 

4.343 

0.000  5.657 

log.a;  =  (logiox)  -^  ill ;  M=  .4342944819 


PART   I 

PLANE   ANALYTIC   GEOMETRY 

CHAPTER   I 
SYSTEMS   OF  COORDINATES 

1.  The  linear  scale.  Analytic  geometry  is  based  upon  a  geo- 
metric representation  of  numbers. 

Choose  a  straight  line  AB  of  indefinite  length  and  upon  it  a 
fixed  point  0.  With  a  convenient  unit  lay  off  the  equal  distances 
0P„  P,P„  PoP„  ...  to  the  right,  and  0Q„  Q,Q„  Q.Q^,  •••to  the 
left.  We  will  now  agree  that  the  points  Pi,  Po,  P3,  •••  shall 
represent  the  positive  integers  1,  2,  3,  •••,  respectively,  and  the 
points  Qi,  Q2,  Qs,  •■'  shall  represent  the  negative  integers  —  1, 
—  2,  —  3,  •••,  respectively. 

-3  -2  -1  1      iH    2  3 

1 1 1 ( 1 1 1 1 ^ 

•i  Q,  Q,  Q,  0  r,     i?     R  P3  B 

Fig.  1 

The  intervals  along  the  line  AB  can  be  divided  into  fractional 
parts  of  the  unit,  thus  obtaining  points  representing  fractional 
numbers.  For  example,  the  point  B  bisecting  the  segment  P1P2 
represents  the  positive  number  1.5. 

The  subdivision  can  be  carried  on  indefinitely  and  we  may 
infer,  finally,  that  the  following  statement  and  its  converse  hold 
concerning  this  representation. 

Each  of  the  points  on  the  line  AB  represents  a  number  ;  namely, 
that  number  lohich  expresses  the  distance  and  direction  of  the  point 
from  0  in  terms  of  the  unit  chosen. 

Conversely,  if  x  is  a  positive  (or  negative)  number,  it  is  repre- 
sented by  a  point  x  units  to  the  right  (or  left)  of  0. 

7 


8  SYSTEMS   OF  COORDINATES  [Chap.  I. 

The  line  AB,  together  with  the  points  constructed  as  explained, 
is  called  a  linear  scale.  It  is  the  geometric  equivalent,  or  graphic 
representation,  of  the  system  of  real  numbers. 

The  point  0  is  called  the  origin ;  it  represents  the  number  zero. 

The  scale  on  a  thermometer  is  an  example  of  a  linear  scale. 
Here  the  points  on  the  scale  represent  the  numbers  expressing 
degrees  of  temperature. 

EXERCISES 

1.  Construct  a  linear  scale  using  half  an  inch  for  the  unit.  On  this  scale, 
mark  the  points  representing  3,  ^,  —  2,  —  2|. 

2.  Is  the  scale  on  an  ordinary  carpenter's  square  a  linear  scale  ?  Where 
is  the  origin  ? 

3.  If  the  origin  of  the  scale  is  moved  two  points  to  the  left,  how  will  this 
affect  the  numbers  represented  by  the  scale  ?  If  the  origin  is  moved  two 
points  to  the  right,  how  will  the  numbers  be  affected  ? 

4.  The  freezing  and  boiling  points  on  a  Fahrenheit  thermometer  are  at  32° 
and  212°  respectively,  while  on  a  centigrade  thermometer  they  are  placed  at  0° 
and  100°.  Compare  the  units  of  these  two  scales.  Five  degrees  below  zero 
on  the  centigrade  is  equivalent  to  what  reading  on  the  Fahrenheit  ?  Con- 
struct the  two  scales  in  this  exercise. 

2.    Directed  segments,  directed  angles.     It 

is  frequently  necessary  to  distinguish  be- 
tween the  two  directions  in  which  a  seg- 
ment may  be  laid  off  on  a  given  straight 
line.     This  is  done  by  calling  one  direction 

positive  and  the  other  negative.     Thus,  if 

^  Fig.  2  4.  n 

we  agree   to  call 

the  direction  from  A  to  B  (Fig.  2)  posi- 
tive, then  we  shall  call  the  direction  from 
jB  to  -4  negative.    Expressed  in  symbols,        \  ^x 

AB^-  BA. 


Segments  of  a  straight  line  to  which  a 
direction  has  been  attached  are  called 
directed  segments. 

In  a  similar  way,  an  angle  can  be 
directed.     For  example,  the  acute  angle 


Arts.  2,3]     DIRECTED  SEGMENTS,   DIRECTED  ANGLES      9 

shown  in  Fig.  3  can  be  described  by  a  line  rotating  with  the 
hands  of  a  clock,  or  clockwise,  from  OB  to  OA ;  or  counterclockwise, 
from  OA  to  OB.  It  is  customary  to  consider  counterclockwise 
rotation  as  positive,  and  clockwise  rotation  as  negative.  Thus,  the 
acute  angle  AOB  is  considered  as  a  positive  angle  and  the  acute 
angle  BOA,  as  a  negative  angle.     In  symbols, 

AOB  =  -  BOA. 

3.  Addition  of  directed  segments,  addition  of  directed  angles.  If 
AB,  BC,  and  AC  are  three  directed  segments  along  the  same  line 
(Fig.  4),  then  the  equation 

AB  +  BC=AC 


Fig.  4 

has  the  following  simple  interpretation ;  namely,  a  point  moving 
along  the  line  from  A  to  B  and  then  from  J5  to  C  is  in  the  same 
final  position  as  it  would  have  been  had  it  moved  directly  from  A 
to  C.  With  this  interpretation,  the  above  equation  is  readily 
seen  to  hold  however  the  points  A,  B,  and  C  are  situated  with 
respect  to  each  other. 
Again,  we  have 

AC-  AB  =  AC+  BA  =  BA  +  AC=  BC. 


Similarly,  the  equation 

AOB  +  BOC=  AOC  (Fig.  5) 

is     interpreted     as     follows ;     rotation 
through     the     angle    AOB     and     then 
through  the  angle  BOC  is  equivalent  to 
rotation  through  the  angle  AOC. 
Again, 

AOC  -  BOC  =  AOC  +  COB  =  AOB. 


Fig.  5 


10  SYSTEMS  OF  COORDINATES  [Chap.  I. 

EXERCISES 

1.  Construct  a  linear  scale,  using  half  an  inch  for  the  unit,  and  mark  the 
points  A,  five  units  to  the  right  of  the  origin,  and  B,  three  units  to  the  left 
of  the  origin.  State  ^the  geometric  meaning  of  OA  —  OB,  of  OB  —  OA,  of 
OA  +  AB.  What  directed  segment  is  equivalent  to  each  ?  What  is  the 
numerical  value  of  each  ?  Is  there  a  directed  segment  in  the  figure  equiva- 
lent to  0A+  OB? 

2.  What  is  the  difference  in  absolute  temperature  between  —  5°  Fahren- 
heit and  20"^  centigrade  ? 

3.  Represent  geometrically  the  difference  in  time  between  10  a.m.  and 
3  P.M.  as  the  difference  between  two  directed  angles. 

4.  In  surveying,  the  azimuth  of  a  line  is  its  direction  expressed  in  degrees, 
measured  from  the  South  point  around  towards  the  West,  or  clockwise. 
Thus  the  azimuth  of  a  line  running  due  North  is  180°  ;  of  a  line  running  due 
East  is  270°  ;  etc. 

What  is  the  azimuth  of  a  line  running  N  25°  E  ?  Of  a  line  running 
N  10°  W? 

5.  What  is  the  difference  in  azimuth  between  two  lines,  one  running 
S  40°  W  and  the  other  S  10°  E  ? 

6.  What  is  the  difference  in  azimuth  between  two  lines,  one  running 
S  40°  E  and  the  other  N  25°  E  ? 

4.  Position  of  a  point  in  a  plane.  When  we  have  once  chosen 
a  unit  of  distance,  one  number  is  sufficient  to  locate  a  point  on  a 
line ;  namely,  the  number  expressing  its  distance  and  direction 
from  a  fixed  point,  the  origin  (Art.  1). 

It  requires  two  numbers  to  locate  a  point  in  a  plane  and  these 
numbers  are  called  the  coordinates  of  the  point. 

The  coordinates  of  a  point  may  be  chosen  in  many  different 
ways.  Any  particular  way  of  choosing  them  gives  rise  to  a 
system  of  coordinates.  There  are  two  systems  of  coordinates  in 
common  use ;  namely,  cartesian  coordinates,  named  after  Rene 
Descartes,  who  first  used  this  system  (1637),  and  polar  coordi- 
nates. These  systems  of  coordinates  will  be  explained  in  the 
succeeding  articles. 

5.  Cartesian  coordinates.  Choose  two  linear  scales  OX  and 
OY  (Fig.  6)  with  their  origins  coinciding  at  0.  Through  any 
point  P  in  the  plane  XOY  draw  parallels  to  OX  and  OY,  meet- 
ing OY  and  OX  in  E  and  D,  respectively.     The  numbers  repre- 


Arts.  4-6] 


RECTANGULAR  COORDINATES 


11 


sented  by  D  and  E  on  their  respective  scales  are  the  cartesian 
coordinates  of  P.  By  article  1,  these  coordinates  express  the  dis- 
tances and  directions  oi  D 
and  E  from  0  in  terms  of  the 
unit  chosen ;  or  the  distances 
and  directions  of  P  from  0  Y 
and  OX  measured  along  par- 
allels to  OX  and  OT,  re- 
spectively. Thus,  if  X  and  y 
denote  the  numbers  repre- 
sented by  the  points  D  and 
E,  respectively,  the  coordi- 
nates of  P  are 

x=OD  =  EP 

and        y  =  OE  —  DP.  Fig.  6 

In  the  same  Avay,  the  coordinates  of  P^  are 

a-i  =  OA  =  E,P^  and  y,  =  OE,  =  AA- 

Hence  any  point  in  the  plane  XO  T  has  an  aj-coordinate  and  a 
^/-coordinate  represented  by  points  on  the  scales  OX  and  OY, 
respectively. 

Conversely,  any  two  numbers  x  and  y  serve  to  locate  a  point 
in  the  plane.  For,  let  D  and  E  be  the  points  representing  x  and 
y  upon  their  respective  scales.  Through  D  draw  a  line  parallel 
to  OY,  and  through  E  a  line  parallel  to  OX.  These  parallels 
meet  in  a  single  point  P ;  the  point  whose  coordinates  are  x  and  y. 

The  scales  OX  and  OY  are  called  the  coordinate  axes,  or  sim- 
ply the  axes.     OX  is  called  the  X-axis,  and  OY  the  F-axis. 

The  segment  OD  is  often  called  the  abscissa  of  the  point  P;  and 
the  segment  DP  the  ordinate  of  P. 

The  units  of  distance  used  in  constructing  the  two  linear  scales 
OX  and  0  Y  are  usually  taken  to  be  the  same,  but  it  is  not  neces- 
sary to  take  them  so.  In  many  cases  it  is  more  convenient  to 
use  different  units. 

6.  Rectangular  coordinates.  The  coordinate  axes  may  intersect 
at  any  angle,  but  it  is  generally  simpler  to  take  them  perpendicu- 


12 


SYSTEMS   OF  COORDINATES 


[Chap.  I. 


lar  to  each  other.     In  this  case,  cartesian  coordinates  are  called 
rectangular  coordinates. 

In  rectangular  coordinates,  the  axes  divide  the  plane  into  four 
quadrants  named  first,  second,  third,  and  fourth  quadrant  as 
indicated  in  Fisr.  7. 


II.  Quadrant 


+  3 


III.  Quadrant 


I.  Quadrant 


rV.  Quadrant 


Fig.  7 


The  algebraic  signs  of   the  coordinates  of   any  point  depend 
upon  the  quadrant  in  which  the  point  lies.     Thus, 


Points  in  Qfadea.vt 

a; 

y 

I 

+ 

+ 

II 

— 

+ 

III 

- 

— 

IV 

+ 

— 

Conversely,  the  algebraic  signs  of  the  coordinates  determine  the 
quadrant  in  which  the  point  lies.  For  example,  if  x=—6  and  ?/=3, 
the  point  lies  in  the  second  quadrant  as  indicated  in  the  figure. 

7.  Notation.  The  notation  P  =  (a,  &),  or  P(a,  &),  indicates  that 
the  coordinates  of  the  point  P  are 

x  =  a  and  y  =  b. 
The  x'-coordinate  is  always  written  first.     For  example,  to  indi- 
cate the  position  of  the  point  P  in  Fig.  7,  we  write  P=(—  5,  3), 
or  P(-  5,  3). 


Akts.  7, 


POLAR  COORDINATES 


13 


EXERCISES 

1.  Draw  the  axes  OX,  OY  and  locate  the  following  points:  (|,  3), 
(2,  -I),  (0,5),  (5,0). 

2.  Where  are  the  points  located  for  which  x  =  0  ?  For  which  x  =  1  ? 
For  which  y  =  0?     For  which  y  =—  1? 

3.  By  means  of  a  geometrical  construction,  locate  accurately  the  points 
( V2,  3),  (VI,  V2),  (\/5,  a/G).     Can  the  point  (0,  w)  be  located  accurately  ? 

4.  The  axes  OX,  OF  are  perpendicular  to  each  other;  locate  the  points 
Pi  =  (l,  2),  P2  =  (5,  5),  and  P3=(5,  2).  Find  the  lengths  of  the  sides  of 
the  triangle  P1P2P3. 

5.  Let  the  axes  OX,  OY  make  an  angle  of  60  degrees  with  each  other  ; 
plot  the  points  in  the  preceding  exercise  and  find  the  lengths  of  the  sides  of 
the  triangle  FiPoPs- 

6.  "With  rectangular  coordinates,  show  that  the  points  (2,  3),  (2,  —  1), 
(—2,  —  1),  and  (—2,  3)  form  a  rectangle.  Find  the  lengths  of  the  sides, 
the  lengths  of  the  diagonals,  and  the  area  of  this  rectangle. 

7.  With  rectangular  coordinates,  show  that  the  points  (1,  1),  (3,  1),  and 
(2,  2)  form  an  isosceles  triangle  which  is  half  a  square.  Find  the  coordi- 
nates of  the  fourth  vertex,  the  lengths  of  the  sides,  the  lengths  of  the  diago- 
nals, and  the  area  of  the  square. 

8.  Polar  coordinates.  The  position  of  a  point  P  in  a  plane  is 
also  determined  by  its  distance  ?*,  in  terms  of  a  given  unit  of  dis- 
tance, from  a  fixed  point  0, 
called  tlie  origin  or  pole,  and 
the  angle  $  which  OP  makes 
with  the  positive  direction  of 
a  fixed  linear  scale  OX,  called 
the  initial  line  or  axis  (Fig.  8). 
OP  is  called  the  radius  vector, 
and  the  angle  XOP  the  vec- 
torial angle,  or  simply  the 
angle,  r  and  6  are  the  polar 
coordinates  of  P.  The  nota- 
tion P  =  (r,  9),  or  P(r,  6),  means  that  r  and  6  are  the  polar  coordi- 
nates of  P. 

A  given  number  r  and  a  given  angle  0  determine  uniquely  the 
position  of  a  point  in  a  plane  with  reference  to  a  fixed  origin  and 
initial  line.    For,  imagine  the  initial  line  OX  (Fig.  9)  to  be  rotated 


Fig.  8 


14 


SYSTEMS  OF   COORDINATES 


[Chap.  I. 


through  the  given  angle  $  about  0  into  the  position  OX'.  On 
OX'  mark  the  point  which  represents  the  given  number  r.  There 
is  but  one  such  point.  Por  example,  the  point  P(-~  5,  —  30°)  is 
obtained  by  rotating  OX  through  the  angle  —  30°  and  marking 
the  point  5  units  from  0  on  the  negative  end  of  the  scale  OX'. 


^\. 


Fig.  9 

On  the  other  hand,  a  given  point  has  many  sets  of  polar  coor- 
dinates. For  example,  the  point  P  in  Fig.  9  is  (—5,  —30°), 
(5,  150°),  (-5,  330°),  (5,  -210°),  etc.  It  is  always  possible, 
however,  and  usually  most  convenient,  to  choose  the  polar  coor- 
dinates of  a  point  so  that  the  radius  vector  shall  be  a  positive 
number,  and  0  <  0^2 it. 

9.  Relation  between  rectangular  coordinates  and  polar  coordinates. 
In  Fig.  10,  let  0  be  the  origin  and  OX  the  initial  line,  so  that 

the  polar  coordinates  of 
any  point  as  P  are  : 


r  =  OF  and 


XOP. 


Fig.  10 


Let  OX  and  OF  be  the 
X-  and  y-axes,  respec- 
tively, so  that  the  rec- 
tangular coordinates  of 
P  are 

X  =  OB  and  y  =  DP. 


Now,  wherever  the  point  P  may  be  located  in  the  plane,  we 

always  have 

oc  =  r  cos  6  and  ?/  =  r  sin  6.  (1) 


Art.  9]  RELATION  BETWEEN   COORDINATES  15 

From  equations  (1),  by  squaring  and  adding,  we  obtain 

a;2  +  2/'  =  r\  (2) 

Also  from  equations  (1),  we  have 


=  arc  cos  -  =  arc  sin  ^  •  (3) 

r  r 


From  (2)  and  (3), 


r  =  ±  Vx2  +  ?/2  and  8  =  arc  cos .  =  arc  sin -^ 

±  Vic2  +  2/2  ^  y/^2  +  yt 

=  arc  tan  — .  (4) 

Equations  (1)  serve  to  change  the  polar  coordinates  of  a  point 
into  rectangular  coordinates  ;  and  equations  (4)  are  used  to  change 
rectangular  coordinates  into  polar  coordinates.  For  example,  the 
rectangular  coordinates  of  the  point  P(—5,  —  30°)  are 

a;  =  -  5cos(-  30°)=~^^^y  =  -5sin  (-  30°)  =  |-    (See  Fig. 9.) 
Again,  the  polar  coordinates  of  the  point  P(—  3,  —  4)  are 


r  =  V9  -I-  16  =  5,  ^  =  arc  cos  (-  f)=  arc  sin  (-  4)  =  233°  8'. 

In  solving  this  problem,  r  was  taken  to  be  the  positive  square 
root  of  25.     With  r  =  —  5, 

6  =  arc  cos  (|)  =  arc  sin  (i )  =  53°  8'. 


EXERCISES 

1.  Plot  the  points    (.3,  -30°),    ^-4,   ^V  (3,  2  radians).     Find  the 
rectangular  coordinates  of  these  points. 

2.  Find  the  polar  coordinates  of  the  points  whose  rectangular  coordi- 
nates are  (3,  —  7),  (4,  2),  (-  3,  —  5).     Plot  the  points. 

3.  Where  do  the  points  lie  for  which  the  radius  vector  is  constant? 
For  which  the  vectorial  angle  is  constant  ? 

4.  If  a;  =  4  and  r  =  5,  find  y  and  0.    Is  there  more  than  one  point  satis- 
fying the  given  conditions  ? 


16 


SYSTEMS   OF   COORDINATES 


[Chap.  I. 


5.  With  a  centigrade  scale  on  the  X-axis  and  a  Fahrenheit  scale  on  the 
F-axis,  plot  a  number  of  points  whose  coordinates  represent  the  same  abso- 
lute temperature.  For  example  (0,  32),  (5,  41),  etc.  Try  to  show  that  all 
these  points  must  lie  on  a  straight  hne,  and  to  find  where  this  straight  line 
meets  the  X-axis. 

6.  Plot  a  number  of  points  for  which  the  radius  vector  is  twice  the 
abscissa.     Join  the  points  plotted.     Do  they  lie  on  a  straight  line  ? 

7.  Elevations  of  points  on  the  ground  above  a  fixed  datum  plane  are 
sometimes  expressed  in  meters,  while  the  distances  of  these  points  from  a 
given  place  of  beginning  may  be  expressed  in  feet. 

Plot  the  points  whose  elevations  and  distances  are  as  follows : 

Distance  Elevation 

100  feet  3.2  meters 

200  feet  6.0  meters 

250  feet  8.0  meters 

300  feet  7.0  meters 

400  feet  5.0  meters 

500  feet  3.0  meters 

600  feet  -2.0  meters 

A  broken  line  drawn  through  the  points  thus  determined  is  called  a 
profile. 

8.  Reduce  the  elevations  of  exercise  7  to  feet  and  plot  the  same  profile. 

9.  From  a  point  0,  the  azimuths  and  distances  to  three  points  A,  B, 
and  C  ai'e  as  follows  : 


Azimuth 

Distance 

A 
B 

C 

120° 
180° 
240° 

10  rds. 
15  rds. 
12  rds. 

Make  an  accurate  map  of  the  triangle  ABC.  With  0  as  origin  and  OB 
as  F-axis,  compute  the  rectangular  coordinates  of  A,  B,  and  C.  With  0  as 
pole  and  OB  as  initial  line,  compute  the  polar  coordinates  of  A,  B,  and  C. 

10.  Construct  a  scale  on  the  X-axis,  the  unit  of  measure  representing 
one  foot ;  and  a  scale  on  the  F-axis,  the  unit  of  measure  representing  one 
meter.  Take  one  meter  equivalent  to  3.28  feet.  Plot  a  number  of  points 
whose  coordinates  represent  the  same  distance,  for  example  (3.28,  1), 
(6.56,  2),  etc.  Show  that  these  points  lie  on  a  sti'aight  line  passing  through 
the  origin. 


Art.  9]  RELATION  BETWEEN  COORDINATES  17 

11.  Construct  a  scale  on  the  X-axis  representing  British  money,  and  a 
scale  on  the  I'-axis  representing  American  money.  Take  £  1  equivalent  to 
§4.87.  Plot  a  number  of  points  whose  coordinates  represent  the  same  value, 
for  example  (1,  4.87),  (2,  9.74),  etc.  Show  that  these  points  lie  on  a  straight 
line  passing  through  the  origin. 

12.  Plot  the  points  whose  polar  coordinates  are  (—6,  20^),  (  —  5,  —315°), 
[  —  4,  -  j ,  (  —  3,  —  ).  Change  the  coordinates  of  these  points  so  that  r 
and  e  shall  be  positive,  and  d  less  than  3G0°. 

13.  Change  the  polar  coordinates  of  the  points  in  exercise  12  to  rectan- 
gular coordinates. 

14.  With  a  convenient  unit,  mark  the  points  U  and  B  on  the  X-axis, 
representing  the  numbers  1  and  &,  respectively.  On  the  T-axis,  mark  a  point 
A,  representing  the  number  a.  Join  A  to  Z7,  and  through  B  draw  a  parallel 
to  A  U,  meeting  the  F-axis  in  C.     Prove  that  C  represents  the  number  ab. 

15.  With  a  convenient  unit,  mark  the  point  U,  on  the  X-axis,  represent- 
ing the  number  1  ;  and  on  the  T-axis  the  points  B  and  A,  representing  the 
numbers  b  and  a,  respectively.  Join  B  to  U,  and  through  ^4  draw  a  parallel 
to  BU,  meeting  the  X-axis  in  the  point  C.     Show  that  G  represents  the 

number  — . 
b 

16.  With  a  convenient  unit,  mark  the  points  U  and  A  on  the  X-axis, 
representing  the  numbers  —  1  and  a,  respectively  (a  being  a  positive  number). 
On  UA  as  diameter,  draw  a  circle  and  prove  that  it  meets  the  F-axis  in  points 
representing  the  numbers  ±  Va.  In  this  way  construct  geometrically  \/2, 
V3,  V5. 


CHAPTER   II 


DIRECTED   SEGMENTS   AND   AREAS   OF  PLANE   FIGURES 

10.    Projections  upon   the    coordinate   axes.     Let   PjP^   be    any 

directed  segment.     Through  Pi(xi,  y^)  and  P-ii^^,  y^)  draw  parallels 
to  the  axes  as  shown  in  Fig.  11.     The  segments  D^Do  and  E^E^, 

thus  determined  upon 
the  axes,  are  called 
the  projections  of  P1P2 
upon  the  X-axis  and 
upon  the  y-axis,  re- 
spectively. The  pro- 
jections themselves 
are  directed  seg- 
ments, and  therefore 
(Art.  3) 


Fig.  11 

j>iD2  =  D^o  +  oi>2  =  -  oi>i  +  on.2 

E^E.^  =  E^O+  OE^  =  -  OE^  +  OE.2 


2/2  -  Vl' 


(1) 


If  we  call  Pi  the  initial  point  and  P^  the  terminal  point,  the 
projection  of  P1P2  upon  the  X-axis  is  found  by  subtracting  the 
x'-coordinate  of  its  initial  point  from  the  x-coordinate  of  its 
terminal  point.  Similarly,  the  projection  upon  the  T'-axis  is 
found  by  subtracting  the  y-coordinate  of  the  initial  point  from 
the  ^/-coordinate  of  the  terminal  point. 

11.  Inclination  and  slope  of  a  directed  segment.  Let  the  coordi- 
nate axes  be  rectangular.  Through  the  initial  point  of  a  directed 
segment  draw  a  line  parallel  to  the  X-axis,  having  its  positive 
direction  the  same  as  that  axis.  The  line  P^D  (Fig.  12)  is  this 
parallel.  The  positive  angle  through  which  it  is  necessary  to 
rotate  this  parallel  to  make  it  coincide  with  the  given  directed 
segment  is  the  inclination  of  the  segment.     Thus  the  angle  DP^P^ 

18 


Art.  11] 


INCLINATION  AND   SLOPE 


19 


(a) 


^D 


^X 


(h) 


is  the  inclination  of  each  of  the  directed  segments  in  Fig.  -12 
The  inclination  may  have  any  value  from  0°  to  360°  inclusive. 

The  tangent  of 
the  inclination  is 
called  the  slope  of  J'2 

the    directed     seg-        y/^ 
ment.    Through  P,,      ^  '-P3|^  ^x 

the  terminal  point 
of  the  segment, 
draw  a  line  paral- 
lel to  the  r^axis 
and  let  it  meet  the 
parallel  to  the  X- 
axis  in  P3.  The 
tangent  of  the 
angle      DP1P2     is 

then    — ^^.      But 

P3P2  is  equivalent 

to  the  projection  of  P^P^  upon  the  F-axis  and  P1P3  is  equivalent 

to  the  projection  upon  the  X-axis.     Hence, 

?/2  -  2/1 


>X 


Fig.  12 


slope  of  F^Po  =  tan  DPiP^ 


OCa   —  OC't 


(2) 


2  ~  •*  1 

For  example,  the  slope  of  the  segment  joining  (—4,  —2)  to  (2,  5) 

is  5 -(-2)  ^7 

2 -(-4)      6- 

Although  reversing   the  direction    of   a   segment   changes   its 
inclination  by  180°,  it  does  not  change  its  slope.     For, 

slope  of  PoPi  =^^^^^=  slope  of  P.P^. 


EXERCISES 

1.  Determine  the  projections,  tlie  inclination,  and  the  slope  of  each  of 
the  following  directed  segments  : 

(«)  (-2,4),  (3,6);  (6)  (-5,7),  (-4,  -2);  (c)  (3,  -2),  (5,6); 
(d)   (-3,2),   (-2,  -3). 

Draw  each  segment. 


20 


DIRECTED  SEGMENTS  AND  AREAS       [Chap.  II. 


2.  If  the  coordinate  axes  make  an  angle  of  60°  with  each  other,  determine 
the  angle  which  the  directed  segment  (2,  1),  (4,  2)  makes  with  each  axis. 

3.  Draw  the  triangle  whose  vertices  are  (1,2),  (5,4),  (2,6),  usino- 
rectangular  coordinates. 

(ff)  Find  the  lengths  of  the  projections  of  the  sides  upon  the  ^-axis. 
What  is  the  sum  of  these  projections  ? 

(6)  Find  the  inclination  of  each  side.  How  can  the  angles  of  the  triangle 
be  found  from  these  inclinations  ? 

4.  Show  that  the  sum  of  the  projections  of  the  sides  of  any  triangle  upon 
either  axis  is  zero,  provided  that  the  sides  be  taken  in  order  around  the 

triangle. 

5.  Fig.  a  represents  a 
railroad  cutting  in  a  side- 
hill.  The  slope  of  the 
natural  surface  is  1  :  4  and 
that  of  the  proposed  cut- 
ting is  1:2.  At  what 
heights  above  the  bottom 
of  the  cut  and  at  what  dis- 
tances out  from  the  center 
line  are  the  points  of  inter- 
section a  and  b  ? 

6.  Fig.  b  is  the  outline 
of  a  roof  truss  of  80-f  t.  span 
and  20-ft.  rise.  The  spaces 
ab,  be,  etc.,  are  equal  and 

the  members  bf,  eg,  and  dh  are  perpendicular  to  the  member  ae.     Calculate 
the  slopes  of  ae,  bf,  fc,  and  ge  with  respect  to  a  horizontal  axis  ak. 

7.  Calculate  the  slopes  of  cf  and  ch  with  respect  to  the  line  ae  taken  as 
the  horizontal  axis. 

12.    The  length  of  a  segment. 

The  problem  to  find  the  dis- 
tance between  two  points  whose 
coordinates  are  given,  that  is, 
the  length  of  the  segment  join- 
ing them,  depends  upon  the 
problem  of  finding  the  length 
of  one  side  of  a  triangle  when 
the  other  two  sides  and  their  in- 
cluded angle  are  given.  Thus, 
with  cartesian  coordinates,  let 


Fig.  13 


Aet.  12]  THE   LENGTH    OF  A   SEGMENT  21 

-Pi(^'i)  yi)i  ^2(^2?  2/2)  be  the  given  points,  and  let  the  angle  XOY 
be  w  (Fig.  13).  Draw  parallels  to  the  axes  through  Pj  and 
P^  forming  the  triangle  PyP^Pf  The  sides  P^P^  and  P<y_P^  are 
known  from  the  given  coordinates  of  P^  and  Po,  and  the  angle 
P^P^P^  =  the  angle  XOY—  w.     Therefore,  by  the  law  of  cosines, 

^^'^  =  JV^nA^^-2JOT^^5^cos«.  (1) 

If  P1P2  is  a  directed  segment,  P1P3  and  P3P2  are  respec- 
tively equivalent  to  its  projections  upon  the  X-  and  F-axes.  In 
terms  of  these  projections,  formula  (1)  becomes 


P,P,'  =  P,P,'  +  P3A  +  2  P1P3  .  P3P2  cos  CO,  (2) 

or  (Art.  10), 

riP'2^  =  (032  -  »^l)^  +  (^2  -  l/l)'^  +  2(iC2  -  OCi)  (iJa  -  ?^i)cos  w.    (3) 

With  rectangular  coordinates,  w  =  90°  and  the  triangle  P1P3P2  is 
right-angled  at  P3.  We  have,  then,  only  to  find  the  length  of 
the  hypotenuse,  having  given  the  other  two  sides. 

With  polar  coordinates,  let  P^  =  {i\,  $^)  and  P2  =  {vo,  62).     In 

the  triangle  P1OP2,  two  sides  and  the  included  angle  are  known, 

hence 

.^1^2  ^  ^^2  ^  ,,^2  _  2  nr..  cos  (02  -  Oj).  (4) 

EXERCISES 

1.  The  angle  between  the  axes  being  45°,  find  the  distance  between  the 
points  (-  3,  -  5)  and  (5,  2). 

2.  Plot  the  points  whose  polar  coordinates  are  (  —  3,  ^j  and  (2,  — ^j 
and  find  the  distance  between  them.  \  /  v  / 

3.  The  rectangular  coordinates  of  Pi  are  (3,  —  2)  and  the  polar  coordi- 
nates of  P2  are  (-  5,  60°).     Find  the  length  of  P1P2. 

4.  The  vertices  of  a  triangle  are  situated  at  the  points  (5,  —  2),  (—4,  7), 
and  (7,  —  3),  in  rectangular  coordinates.     Find  the  lengths  of  the  sides. 

5.  Milwaukee  is  80  miles  east  of  Madison  and  80  miles  north  of  Chicago. 
What  are  the  polar  coordinates  of  Chicago  with  respect  to  Madison  as  origin 
and  the  line  from  Madison  to  Milwaukee  as  axis  ?     The  polar  coordinates  of 

Portage  being  (40,  —  ] ,  find  the  distance  from  Chicago  to  Portage. 

6.  Show  that  the  formula  (3)  Art.  12,  holds  for  all  positions  of  the  points 
Pi  and  P2. 


22 


DIRECTED   SEGMENTS  AND  AREAS       [Chap.  II. 


13.    Angle  which  one  segment  makes  with  another.     Let  the  seg- 
ments P1P2  ^nd  Q1Q2,  produced  if  necessary,  meet  in  the  point  A 

(Fig.  14).  The  angle  which 
Q1Q2  makes  with  P^P^  is  de- 
fined as  the  positive  angle 
through  tvhich  it  is  necessary 
to  rotate  P1P2  about  A  until 
it  coincides  in  direction  loith 
QiQz-  In  the  figure,  this 
angle  is  P-^AQ^  which  is 
clearly  the  difference  be- 
tween the  inclinations  of  the 
two  segments.  If  O^  a'^d  6^ 
are  respectively  the  inclinations  of  QiQo  ^i^d  P^P^  and  ^  is  the 
angle  P,^Q,  then  ^^6,-6,.  (1) 

Formula  (1)  holds  only  when  0^.  >  ^i-     When  62  <  9y,  the  angle 
which  Q1Q2  makes  with  P1P2  is  given  by  the  formula 


as  the  student  may  easily  verify.     In  either  case 

tan  ^9  —  tan  6, 


tan  </)  =  tan  (^2  —  ^1)  = 


1  +  tan  62  tan  9i 


(2) 


(3) 


If  mj  and  mj  are  respectively  the 
slopes  of  Q1Q2  and  P1P2,  formula  (3) 
becomes 

»W2  —  vrix 


tan  ^  — 


1  -I-  mi,fn\ 


(4) 


As  an  example  of  the  use  of  formula 
(4),  we  will  find  the  angle  which  the  seg- 
ment joining  (3, 5)  to  (—  2,  —  6)  makes 
with  the  segment  joining  (—1,  2)  to 
(3,  —  4)  (Fig.  15).  Here  m,,  the  slope 
of  Q1Q2,  is  equal  to  —  f  and  m-^,  the  slope 
of  P1P2,  is  equal  to  U-.     Hence 


tan  <^  = 


1    S3 

^      10 


-,and<i  =  58°8'. 
23  ^ 


Fig.  If) 


Arts.  14,  15] 


PARALLEL  SEGMENTS 


23 


14.  Parallel  segments.  Parallel  segments  either  have  equal 
inclinations,  as  at  (a)  (Fig.  16),  or  else  their  inclinations  diifer 
by  180°   as   at  (&).     In   either  case,   their   slopes  are  the  same 


Y  P, 


->.Y 


(a) 


Fig.  16 


->X 


(h) 


(Art.  11).  For  example,  the  segment  joining  (1,  2)  to  (—2,  —3) 
is  parallel  to  the  segment  joining  (2,  —1)  to  (o,  4),.  since  the 
slope  of  each  is  f . 

15.  Perpendicular  segments.  When  two  segments  are  perpen- 
dicular to  each  other,  their  inclinations  differ  by  an  odd  multiple 
of  90°  and  therefore,  in  every  case, 

1 


tan  62  =  —  cot  61  = 


tan  ^1' 


'tn.^  =  — 


in^ 


(1) 


Thus,  the  slope  of  each  segment  is  the  negative  reciprocal  of 
the  slope  of  the  other. 

Conversely,  if  the  jjroduct  of  the  slopes  of  two  segments  is  —  1, 
the  segments  are  perpendicular  to  each  other.  For  then  the  tangent 
of  the  inclination  of  one  of  them 
is  equal  to  the  negative  of  the 
cotangent  of  the  inclination  of 
the  other.  Hence,  their  inclina- 
tions differ  by  an  odd  multiple 
of  90° ;  that  is,  the  segments 
are  perpendicular  to  each  other. 

In   Fig.   17,  PjPo   niakes    an 
angle  of  90°  with  Q1Q2,  but  Q1Q2  makes  an  angle  of  270°  with 


r 

f. 

Q.T^ 

y 

1 

s-H 

k 

^«, 

^   -v 

0 

Fig.  17 


24 


DIRECTED   SEGMENTS  AND   AREAS       [Chap.  II. 


EXERCISES 

1.  Find  the  angle  which  the  segment  (—3,  2),  (4,  —  1)  makes  with  the 
segment  (  —  3,  2),  (8,  5).    Draw  the  figure. 

2.  Compute  the  lengths  of  the  sides  and  the  angles  of  the  triangle  whose 
vertices  are  (—3,  2),  (4,  —  1),  and  (8,  5).     Draw  the  figure. 

3.  Show  that  the  triangle  whose  vertices  are  (3,  2),  (0,  3.5),  and  (1,  5.5) 
is  right-angled. 

4.  Show  that  the  segments  (—3,  5),  (3,  2)  and  (—1,  6),  (3,  4)  are 
parallel.  Draw  the  figure  and  compute  the  perpendicular  distance  between 
the  segments. 

5.  Join  the  extremities  of  the  segments  in  the  preceding  exercise  and 
compute  the  area  of  the  quadrilateral  so  formed. 

6.  Draw  the  diagonals  of  the  quadrilateral  in  the  preceding  exercise  and 
find  the  acute  angle  which  one  makes  with  the  other. 


J- 

^2 

E 

^ 

r^\ 

p 

^--^ 

Pi 

0 

1 

\  i 

)  1 

\ 

Fig.  18 


and 

whence 


oc  =  OD  =  OD^  +  J>iD  =  xj  + 


y  =  OE  =  OE^  +  E^E  =  y^  + 


16.  Point  bisecting  a  given 
segment.  Let  Pi  =  (a^i,  y^  and 
Po  =  {x^,  yz) ;  it  is  required  to 
find  the  coordinates  of  the  point 
P=(x,  y)  bisecting  the  segment 
PiPo  (Fig.  18).  The  parallels  to 
the  axes  through  P  must  bisect 
the  projections  AA  and  AA 
in  D  and  E,  respectively.  Hence 
(Art.  10) 


2       ' 
2/2  -  Pi 


OCi  +  ijC2 


2/i  +  2/2 


(1) 


For  example,  the  coordinates  of  the  point  bisecting  the  segment 
(1,3),  (-3,  -l)are 


2 


=  -1, 


y  = 


=  1. 


Art.  17]       POINT   DIVIDING  A   GIVEN   SEGMENT 


25 


17.   Point  dividing  a  given  segment  in  a  given  ratio.    The  results 
of  the  preceding  article  can  be  generalized.     Thus,  suppose  the 
point  P  (Fig.   19)    divides 
the  segment  P1P2  so  that 

PP. 

Then  the  points  D  and  E 
divide  the  projections  of 
PiPo  in  the  same  ratio. 
Hence, 

P,P^D,D^{x-x,)^^, 
PP^      DD,      {x,-x)       ' 


Y 

P2 

^^ 

i: 

p 

^^-^ 

^: 

A 

0 

B,                           D 

J).      x 

and 


PR 


E,E_(y~y,)^^, 


_     EE,      {y,  ~  y) 
Solving  these  equations  for  x  and  y,  we  have 


y^ 


(1) 


For  example,  to  find  the  coordinates  of  the  point  dividing  the 
segment  Pi  =  (2,  4),  P,  =  (-  3,  -  2)  in  the  ratio  2  :  3,  we  have 
r  =  f .  Substituting  in  the  above  formulas  we  find  « =  0  and 
y  =  \.  Hence  the  point  (0,  f)  divides  the  given  segment  in  the 
ratio  2  :  3. 

EXERCISES 

1.  Find  the  coordinates  of  the  points  which  bisect  the  sides  of  the  tri- 
angle (2,  5),  (-2,2),  (4,  -5). 

2.  In  the  preceding  exercise,  join  the  vertices  to  the  mid-points  of  the 
sides  opposite  and  show  that  the  points  dividing  each  segment  from  vertex 
to  opposite  side  in  the  ratio  2  :  1  coincide. 

3.  Generalize  the  preceding  exercise  and  thus  prove  that  the  medians  of 
any  triangle  meet  in  a  point. 

4.  Show  that  the  points  (2,  3),  (4,  1),  (8,  2)  and  (6,  4)  form  a  parallelo- 
gram.    Find  the  coordinates  of  the  mid-points  of  the  diagonals. 

5.  Find  the  coordinates  of  the  points  which  trisect  the  segment  (6,  4), 
(-3,1). 


26 


DIRECTED   SEGMENTS  AND   AREAS       [Chap.  II. 


6.  Find  the  coordinates  of  the  point  P  dividing  the  segment  Pi  =  (.3,  4), 
Fi  =  (—  2,  —  6)  in  the  ratio  3  : 5.  Prove  the  result  by  calculating  the  lengths 
of  the  segments  PiP,  PP-^  and  showing  that  their  ratio  is  |. 

7.  The  segment  in  the  preceding  ex- 
ercise crosses  both  axes.  Find  the  co- 
ordinates of  the  points  of  crossing. 


1 8.  Area  of  a  triangle,  one  vertex 
at  the  origin.  To  find  the  area  of 
the  triangle  OP^P.,  (Fig.  20),  let 

^1=  ('^'i,  Vi),  P2^{^'2,  Vi),  and  A  A, 
the  projection  of  P1P2  upon  the 
a;-axis.  Then,  if  A  represents  the 
required  area. 


Fig.  20 


A  =  trapezoid  P^P^D^D^  +  triangle  P^OD^  —  triangle  OD^P^ 

Ml 
2 


^  (Vi  +  JhX^i  -  -^2)  I  X2y2 
2  2 


Hence,  we  have 


(^1^2  -  ^2^1) 


(1) 


The  expression  x^^y^  —  %Vi  is  a  determinant  and  is  often  written 
thus; 

X,     y^ 

In  determinant  notation,  the  formula  for  the  area  of  the  triangle 
is  then 


2/1 

^2      V'l 


(2) 


For  example,  the  area  of  the  triangle  formed  by  joining  the 
extremities  of  the  segment  Pj  =  (3,  1),  P2  =  (1?  3)  to  the  origin  is 


3     1 
1     3 


=  4. 


19.  Sign  of  the  expression  {oc\V^2  —  *2?/i)-  The  sign  of  the 
expression  {x-^y^  —  x^yi)  is  not  the  same  for  all  positions  of  the 
segment  P1P2.  Thus,  if  Pj  =  (3,  1)  and  P^  =  (1,  3),  the  expres- 
sion has  the  value   -\-  8,  Avhile  for  the   segment  P^  =  (1,    —  2), 


Art.  19]         SIGN   OF   THE   EXPRESSION    ixiy2  -  2/2X1) 


27 


P2  =  (  — 1,  1),  which  has  the  same  length  and  the  same  slope  as 
the  "former,  the  expression  (xiy2  —  x^yy)  has  the  value  —  1. 
Changing  to  polar  coordinates  by  means  of  the  relations 

Xi  =■  i\  cos  61,  2/1=  r^  sin  ^1,  x^  =  r^  cos  02,  y-i  =  r^,  sin  B^  (Art.  9), 
the  expression  {x-^y^  —  ^^\)  becomes 

ri9-2(cos  di  sin  62  —  cos  62  sin  ^1)  =  r{i\  sin  (82— Oi). 

Since  r^  and  7*2  may  be  considered  as  positive  numbers  (Art.  8), 
the  sign  of  (x^y2  —  x^yi)  will  be  positive  when  sin  (^2  —  ^1)  is  posi- 


FiG.  21 


tive  ;  that  is,  when  62  —  &i  is  an  angle  in  the  first,  or  the  second, 
quadrant.  In  either  case,  the  segment  PiP^  has  a  position  such 
that,  in  passing  from  Pi  to  P2>  the  origin  lies  to  the  left  as  at  (a). 
Fig  21. 

On  the  other  hand,  the  expression  {x{y2  —  x^^  will  be  negative 
when  ^2  —  ^1  is  ^"^  angle  in  the  third,  or  the  fourth,  quadrant ;  and 
then  the  segment  P1P2  has  a  position  such  that,  in  passing  from 
Pi  to  P2,  the  origin  lies  to  the  right  as  at  (&). 

Conversely,  if  the  segment  P1P2  has  a  position  such  that  the 
origin  lies  to  the  left  (or  the  right)  when  the  segment  is  traversed 
from  Pi  to  P2,  the  sign  of  {Xyy2  —  X2yi)  will  be  positive  (or  nega- 
tive). For  then  the  angle  O2  —  &i  must  lie  in  the  first,  or  the 
second,  quadrant  (or  in  the  third,  or  the  fourth,  quadrant).     Con- 


28 


DIRECTED   SEGMENTS  AND   AREAS      [Chap.  II. 


sequently  the  area  of  the  triangle  OPyP^,  which  is  \  (.x*i?/2  —  ^iVx), 
is  positive  when  the  origin  lies  to  the  left,  as  at  (a),  Fig.  21,  and 
negative  when  the  origin  lies  to  the  right,  as  at  (h). 


EXERCISES 

1.  P\  =  (5,  3)  and  P2  =  (—  1,  —  3) ;  determine  the  area  of  OP1P2,  0  being 
the  origin.     Explain  the  sign  of  the  result.     Draw  the  figure. 

2.  If  0  is  the  pole,  show  that  the  area  of  the  triangle  OP1P2  is 

1  rir2  sin  (6I2  -  di)-, 
where  Pi  =  (n,  Oi)  and  P2=  (r2,  dt). 

3.  If  Pi  =[5,  -"j  and  P2=  (3,  -  30°),  find  the  area  of  OP1P2. 

4.  Given  Pi  =  (3,  -  60°)  and  P2=  (3,  4),  find  the  area  of  OP1P2. 

5.  When  the  segment  P1P2  passes  through  the  origin,  what  is  the  value  of 
the  expression. (;i-i2/2  —  X2J/1)  ? 

6.  If  Pi  =  (— 3,   1)  and  P2  =  (l,  —2),  in  which  quadrant  is  the  angle 
^2  —  ^1  ?     Draw  the  figure  and  find  the  area  of  OP1P2. 


Fig.  22 

20.  Area  of  a  triangle,  vertices  in  any  position.  Join  the  ver- 
tices of  the  triangle  to  the  origin  0.  Let  P^,  P^,  P^  (Fig.  22)  be 
the  vertices,  taken  in  counterclockwise  order  about  the  triangle. 
The  area  of  P^P^P^  is  then  given  by  the  formula 

area  PiP-zP^  =  area  OP1P2  +  area  OPiP^  +  area  OP3P1.      (1) 


Art.  20]  AREA   OF  A   TRIANGLE  29 

Thus  in  (a),  each  of  the  component  triangles  has  a  positive  area, 
by  the  preceding  article,  and  their  sum  is  obviously  the  area  of 
F^P^P^.  In  (&),  the  areas  of  OP^Pz  and  OP^P^  are  positive  num- 
bers, while  the  area  of  OP^Pi  is  a  negative  number.  The  alge- 
bl-aic  sum  of  these  numbers  is  clearly  the  area  of  P^^P^Pz-  Finally, 
in  (c),  the  axea  of  OPxP-i  is  a  positive  number,  while  the  areas  of 
the  remaining  two  triangles  are  expressed  by  negative  numbers. 
The  algebraic  sum  of  these  numbers  is  again  the  area  of  P^P^P^. 
Replacing  the  area  of  each  component  triangle  in  (1)  by  its 
value  in  terms  of  the  coordinates  of  the  vertices  (Art.  18),  we  have 

area  PiP->Ps=l[(^iU-2-'X-2Ui)  +  (JC'iUi--^s!/i)  +  (xsUi-^iUs}]-    (2) 


The  area  can  be  expressed  in  determinant  notation.     Thus 

(3) 


\^l    2/1     1 
area  jPi PoPs  =-\^2    Vi    1 


\X'A     2/3      1 

since,  if  the  determinant  is  expanded,  the  result  agrees  with 
formula  (2). 

The  following  is  a  convenient  rule  for  computing  the  area. 

Let  P1P2P3  be  the  vertices,  taken  in  counterclockwise  order  about  the 
triangle.     Arrange  the  coordinates  in  rows,  thus 


Vv     2/2    Vz    Vi 

multij)ly  each  x  by  the  y  standing  in  the  next  column  to  the  right  and 
add  the  products,  thus 

^iVi  +  X2I/3  +  -Ml ; 

multiply  each  y  by  the  x  in  the  next  column  to  the  right  and  add  the 
products,  thus 

y^Xo  +  ?/o>-3  +  ^/3.^•l ; 

subtract  the  latter  sum  from  the  former  and  take  hcdf  the  difference, 
the  residt  is  the  area  of  the  triangle  PjPo-fs- 

For    example,    to    find    the    area    of    the   triangle   whose  vertices    are 
Pi  =  (-  1,  3),  P2=(3,  2),  P3  =  (5,  4)  (Fig.  23),  arrange  the  coordinates  as 


80 


DIRECTED   SEGMENTS  AND  AREAS     [Chap.  II. 


Fig.  23 


J_  _  ___^  . ,^r^^     . -i£ ^ 1 . 1 

in  the  rule,  being  careful  to  note 
that  the  vertices  are  taken  in 
counterclockwise  order ;  thus 

-  1     3     5     -  1 
3    2     4         3. 

The  area  is  then 

1  [(_  2 -f  12  +  15) 

_(9  +  10-4)]=5. 


If    the  vertices  are  taken  in 
clockwise    order    about  the   tri- 
angle, the  result  obtained  by  using  formula  (2)  or  formula  (3)  or  the  rule 
just  stated  will  be  numerically  the  same  but  will  be  negative  in  sign,  as  the 
student  may  easily  verify. 


6,  ^ 
'  3 


6, 


Draw  the  figure. 


EXERCISES 

1.  Find  the  area  of  the  triangle  whose  vertices  are  (2,  —  6),  (—  9,  7), 
(8,  3). 

2.  Find  the  area  of  the  triangle  whose  vertices  are  (—1,  —2),  (2,  1), 
and  (3,  2).     Explain  the  result. 

3.  Find  the  area  of  the  triangle  whose  vertices  in  polar  coordinates  are 

'¥)•  ^"^  {''  T 

4.  The  vertices  of  a  quadrilateral  are  (—  1,  6),  (8,  10),  (10,  —2),  and 
(—5,  —  8).  Compute  the  area  of  the  quadrilateral  by  dividing  it  into  two 
triangles.     Draw  the  figure. 

5.  When  three  points  are  in  the  same  straight  line,  they  are  said  to  be 
collinear.     Show  that  the  points  (1,  3),  (3,  1),  and  (4,  0)  are  coUinear. 

6.  Where  will  the  line  joining  the  points  (2,  5)  and  (3,  6)  meet  the  axes  ? 

7.  If  Pi{xi,  2/i),  P2{x2,  2/2),  and  P3(xs,  2/3)  are  three  collinear  points, 
show  that 


=  0. 


State  and  prove  the  converse. 

21.  Area  of  any  polygon.  Formula  (2)  of  article  20  can  be 
extended  to  find  the  area  of  any  polygon  when  the  coordinates  of 
the  vertices  are  given.  Thus  when  the  vertices,  taken  in  counter- 
clockwise order  about  the  polygon,  are  joined  to  the  origin,  a 


Xl      111 

1 

X2      2/1 

1 

X3       2/3 

1 

Art.  21] 


AREA  OF  ANY   POLYGON 


31 


1 

']Z  ^         ;                  -       "i^ 

^'    2    ~       S 

^'      ^               S^ 

..o,^Z__7    .    _           S,.    _      . 

iSiS      ^l.                      5 

2        g      "                         S 

_    .      ^Z   ^    j'      _               -    S^-  , 

/      ''        S    C-.                      5  ■' 

?    ^      ^?:      : :=*£ 

2   ^f-.    ^t.    .           _         ,.-! 

4      Z      ^"                        ^--^ 

Z  2   ,'   ,^i!..      _"    " 

t^    ,  — ;--: 

__,^','      , --     _ 

V ■'    -^' 

; iziiZ": : i_ 

^    '                 ~^f-3       ^t-      Jia                   '^^          X 

-    -    -                        s       -.4-        ^                      1            .i 

__         _            X                                 _                 ' 

...    ± -    _. 

Fig.  24 


number  of  component 
triangles  are  formed 
(Fig.  24).  It  is  geo- 
metrically evident  that 
the  algebraic  sum  of 
the  areas  of  these  tri- 
angles is  the  area  of 
the  polygon.  A  con- 
venient rule  for  com- 
puting the  area  of  a 
polygon  is,  therefore, 
obtained  by  extending 

the  rule  in  Art.  20.     Thus,  write  the  x's  over  the  ?/'s  and  form  the 
cross-products : 

1  2  3        **^4    * "  •    to  *^'1 ) 

Vl        2/2       ^3       2/4    •••   Vn       VV 

The  required  area  is  then 

(1) 

For  example,  to  find  the  area  of  the  quadrilateral  whose  vertices  are,  in 

counterclockwise  order  (8,  10),   (-1,  6),  (-5,  -8),  and  (10,  -2),  we 

have 

8     _  1     -  .5       10       8 

10         6     -8     -2     10 
and  the  area  is,  therefore, 

1  [(48  +  8  +  10  +  100)  -  (  -  10  -  30  -  80  -  16)]  =  151. 


EXERCISES 

1.  The  vertices  of  a  hexagon  are  (6,  1),  (.3,  —10),  (—3,  —5),  (—12, 
0),  (—  4,  6),  and  (9,  —  4).    Draw  the  hexagon  and  compute  its  area. 

2.  A  surveyor  finds  that  the  corners  of  a  four-sided  field  are  situated, 
with  respect  to  a  north  and  south  road  and  an  east  and  west  road,  as 
follows:  ^  =  (25,  32),  £=(48,  65),  C=(94,  -10),  and  D  =  (30,  -40). 
Distances  are  measured  in  rods.  Make  a  map  of  the  field  and  compute  the 
number  of  acres  it  contains.     (160  square  rods  =  1  acre). 

3.  From  a  point  0  in  a  quadrangular  field,  the  distances  and  directions  to 
the  corners  are  as  follows  :  J[=  120  feet,  N.  65°  E.  ;  J5  =  216  feet,  N.  32°  W. 
C= 320  feet,  S.  74°  W. ;  2)  =  65  feet,  S.  23°  E.     Make  a  map  of  the  field  and 
compute  its  area. 


32 


DIRECTED   SEGMENTS  AND   AREAS       [Chap.  IL 


4.  In  surveying,  points  are  frequently  located  by  azimuth  and  distance 
from  a  given  point,  tliat  is,  by  polar  coordinates.  It  frequently  becomes 
necessary  to  plot  the  outlines  of  tracts  of  land  determined  in  this  way  and  to 
calculate  areas. 

Dravs^  the  polygonal  figure  whose  vertices  are  determined  by  the  following 
azimuths  and  distances : 


AZLMUTH 

DiSTAXCE 

Azimuth 

Distance 

125° 

115  feet 

342° 

175  feet 

170° 

160  feet 

15° 

40  feet 

250° 

200  feet 

73° 

10  feet 

(a)  Calculate  the  coordinates  of  the  vertices  of  this  figure  referred  to  N. 
and  S.  and  E.  and  W.  lines  through  the  given  fixed  point,  as  origin. 

(b)  Calculate  the  directions  of  the  sides  of  this  figure. 

(c)  Calculate  the  area  of  the  polygon. 

5.  Fig.  24  A  represents  a  cross  section  of  one  side  of  a  railroad  cutting. 
Calculate  the  area  of  this  section,  using  coordinates  as  shown. 

In  railroad  field  books  the  data  for  this 
problem  would  generally  be  recorded  as 
follows : 

Center 

\  i2\,.--'9\/'ii\/^\/'o  ; 

]  0 /\5/\l2/\20/'\8  \ 
The  ordinate,  or  depth  of  cutting,  is  writ- 
ten above  and  the  distance  out  from  the 
center  line  (abscissa)  is  written  below. 
The  coordinates  (|)  are  not  actually  re- 
corded as  the  number  8  is  the  fixed  width 
of  the  bottom  of  the  cut.  Arranging  the 
coordinates  in  the  above  manner,  the  correct  result  is  obtained  by  taking 
positive  products  along  diagonal  lines  sloping  downwards  towards  the  right 
(shown  by  full  lines)  and  negative  products  along  the  other  diagonals  (shown 
by  dotted  lines) . 

6.  Compute  the  area  of  cross  section  given  by  the  following  cross  section 
notes,  left  and  right  of  the  center  line : 

Center 

fO\     _4      _6_     a     1_0    J_Q    12     18.     16     (0\ 
IvS/      1?    iO     5    '0         5       I'S    ^5     32     \S/ 

What  side  slope  of  the  finished  cut  has  been  assumed  in  this  problem  ? 

7.  Drop  perpendiculars  from  the  vertices  of  a  polygon  upon  the  X-axis, 
as  in  Fig.  24.  Show  that  the  area  of  the  polygon  is  the  algebraic  sum  of  the 
areas  of  the  trapezoids  thus  formed.  Compute  the  area  of  the  hexagon  in 
exercise  1  by  this  method. 


Fig.  24  a 


CHAPTER   III 
FUNCTIONS   AND  THEIR   GRAPHIC  REPRESENTATION 

22.  Constants  and  variables.  The  numbers  and  magnitudes 
considered  in  mathematics  are  either  constants  or  variables.  The 
coordinates  of  a  fixed  point  are  constants ;  the  coordinates  of  a 
moving  point  are  variables. 

23.  Functions.  If  to  each  given  value  of  a  variable  x  there  corre- 
spond one  or  more  values  of  a  variable  y,  then  y  is  called  a  function 
of  X. 

As  examples,  the  cost  of  a  money  order  is  a  function  of  the 
amount;  the  temperature  at  a  given  place  is  a  function  of  the 
time;  the  cost  of  insurance  is  a  function  of  the  age  of  the  in- 
sured ;  the  distance  a  body  falls  freely  in  space  is  a  function  of 
the  time  the  body  has  been  falling. 

24.  Notation.  To  denote  that  ?/  is  a  function  of  x,  the  notation 
y  =f{x)  (read  y  equals  /  of  x)  is  used. 

When  several  functions  are  to  be  considered  in  the  same  prob- 
lem, different  symbols  are  used.  Thus,  y=f(x),  y=f-,(x),  .  .  . 
(read  y  equals  f^  of  x,  y  equals  fo  of  x,  .  .  .).  Or  use  is  made  of 
Greek  letters,  as  y  =  <^(.v),  y  =  ^(x),  •  •  •  (read  y  equals  phi  of 
X,  y  equals  psi  of  x,  .  .  .). 

25.  Determination  of  functional  correspondence.  A  functional 
correspondence  can  be-  established,  or  set  up,  between  two  variables 
in  different  ways.  Thus,  the  correspondence  may  be  primarily 
established : 

I.  By  an  equation  connecting  the  two  variables,  as  y  =  x^; 

II.  By  a  table  exhibiting  corresponding  values  of  the  variables, 
as  a  table  of  logarithms ; 

III.  By  a  curve  drawn  automatically,  thus  exhibiting  graphi- 
cally the  correspondence  between  two  variables. 

33 


34  GRAPHIC   REPRESENTATION  [Chap.  III. 

26.  Dependent  and  independent  variables.  If  functional  corre- 
spondence is  established  by  au  equation,  the  value  (or  values)  of 
the  function  can,  in  general  be  readily  compiited  for  any  value 
assigned  to  the  variable  x.  Thus,  for  example,  if  y  =  2a^,  the 
value  of  y  is  easily  computed  for  any  assigned  value  of  x.  In 
general,  y  (the  function)  is  called  the  dependent  variable,  and  x, 
the  independent  variable. 

27.  Graphic  representation.  A  table  of  corresponding  values 
of  a  function  and  the  independent  variable  can  be  derived  from 
the  equation  by  assigning  to  the  independent  variable  a  series 
of  values,  arbitrarily  chosen,  and  computing  the  corresponding 
values  of  the  function.  With  these  corresponding  values  of  x 
and  y  as  rectangular  coordinates,  a  series  of  points  can  be  con- 
structed. The  ordinates  of  these  points,  taken  together,  form  a 
graphic  representation  of  the  function.  For  the  functions  con- 
sidered in  this  book,  a  curve  can  be  drawn  through  all  the  points 
constructed  as  above.  This  curve  is  called  the  graph  of  the  function. 
Thus,  from  the  equation  y  =  2x^,  we  obtain  the  following  table  of 
corresponding  values : 

x  =  -3,  -2,  -1,  0,  1,  2,     3,     4, 

y=   18,       8,       2,  0,  2,  8,  18,  32, 

The  process  of  constructing  the  points  whose  coordinates  are 
given  in  the  table  and  drawing  the  curve  through  them,  is  called 
plotting.  Figure  25  shows  the  completed  graph.  In  constructing 
this  graph,  the  unit  of  the  scale  on  the  X-axis  was  taken  four 
times  as  great  as  the  unit  on  the  Y'-axis  in  order  to  represent 
more  of  the  curve  within  a  small  compass  (cf.  Art.  5).  The  curve 
shows  at  a  glance  the  change  in  value  of  the  function  for  any 
given  change  in  value  of  the  independent  variable  x.  For  exam- 
ple, as  X  changes  from  —  3  to  -f  3,  the  point  which  represents  it 
moves  from  D^  to  D^.  At  the  same  time  the  function  y  first 
decreases  from  18  to  0  and  then  increases  from  0  to  18. 

When  a  function  ceases  to  decrease  and  begins  to  increase,  or 
vice  versa,  it  is  said  to  have  a  turning  point.  Thus,  the  function 
y  =  2  x^  has  a  turning  point  at  the  origin. 


Art.  27] 


GRAPHIC   REPRESENTATION 


35 


When  a  function  has  no  turning  points,  it  is  called  a  monotone 
function. 

It  is  important  to  know  whether  a  function  has  turning  points 
or  not,  and,  if  it  has,  to  know  for  what  values  of  the  independent 


Fig.  25 


variable  they  exist.     At  a  turning  point,  the  function  has  a  max- 
imum or  a  minimum  value. 


EXERCISES 

1.  Draw  the  graph  of  the  function  y  =  2  x  +  3.  Find  the  coordinates  of 
the  points  where  the  graph  crosses  the  axes.  Does  the  function  have  turn- 
ing points,  or  is  it  a  monotone  function  ? 

2.  Draw  the  graph  of  the  function  expressing  the  law  of  falling  bodies, 
s  =  l  gt^.  Take  g  =  32  and  corresponding  values  of  s  and  t  as  ordinates  and 
abscissas  respectively.  Make  the  unit  of  the  scale  on  the  f-axis  ten  times  as 
great  as  the  unit  on  the  s-axis.     Where  is  the  turning  point  of  the  function  s  ? 

3.  Draw  the  graph  of  the  equation  expressing  Boyle's  law,  pv  =  k.  Take 
A;  =  4  and  corresponding  values  of  p  and  v  as  ordinates  and  abscissas  respec- 
tively. Make  the  units  the  same  on  each  axis.  Is  ^j  a  monotone  function 
of  V  or  not  ? 

4.  Make  careful  drawings  of  the  graphs  of  the  function  ?/  =  a:"  for  n=—1, 
0,  1,2,  and  3.  Use  the  same  axes  and  preserve  the  figures.  For  which  of 
the  given  values  of  n  is  y  not  a  monotone  function  of  a;  ? 


36 


GRAPHIC   REPRESENTATION 


[Chap.  III. 


5.  Draw  the  graph  of  the  function  y  =  4:  —  x^.  Determine  the  value  of 
X  for  which  this  function  has  a  turning  point.  Has  the  function  a  maximum 
or  a  minimum  value  at  the  turning  point  ? 

6.  Draw  the  graph  of  the  function  y  =  x^  —  ix  +  S.  Determine  the  coor- 
dinates of  the  turning  point.  Has  the  function  a  maximum  or  a  minimum 
value  at  the  turning  point  ? 

28.  Single-valued  and  multiple-valued  functions.  When  there 
corresponds  but  one  value  of  the  function  to  each  given  value  of 
the  independent  variable,  the  function  is  called  single-valued.  If 
there  is  more  than  one  value  of  the  function  corresponding  to  any 
given  value  of  the  variable,  the  function  is  called  multiple-valued. 
For  example,  the  function  y  =  2  x^  is  single-valued  ;  but  the  func- 
tion y^  =  2x  is  multiple-valued,  since  to  each  value  of  x  there 
correspond  two  values  of  ?/. 

The  following  is  a  table  of  corresponding  values  for  the  func- 
tion y^  =  2x. 

x^      -2,      -1,  0,  1,       2,  3,  4,  .... 

2/  =  Imag.,  Imag.,  0,    ±V2,    ±2,    ±  a/6,    ±  V8,   ..-. 


_       -            -            _p 

TpL       "         "                                            it 

II      III      1     — — " " 

-r''"^ 

^--  ^ 

^^ 

^n. 

;? 

^  ^ 

7 

z     ~         ~ 

•  1                                                                                                          " 

"    1  '.T        !  '   1                                                                                    ■'       i 

^   <\  '                                                                                                                                                                      T 

1    !           1   X 

^        -_             j^          - 

S.  _            ±     .        . 

■*,, 

_ _■=;                          "                                                _  ^     _      _ 

-  ^,^ 

^  S  j^ 

^.■.... 

■*  .*J^ 

"""■.-                                                         ' 

Fig.  26 


Arts.  28,  29] 


SYMMETRY 


37 


The  graph  is  shown  in  Fig.  26,  where  the  same  unit  is  used  for 
the  scale  on  the  I'-axis  as  for  the  scale  on  the  X-axis. 
The  curves  in  Figs.  25  and  26  are  called  parabolas. 

29.  Symmetry.  A  curve,  or  graph,  is  symmetrical  -with  respect 
to  a  straight  line  when  the  line  bisects  all  the  chords  of  the  curve 
drawn  perpendicidar  to  it. 

For  example,  the  parabola  shown  in  Fig.  25  is  symmetrical 
with  res]3ect  to  the  I^axis  and  the  j)arabola  in  Fig.  26  is  sym- 
metrical with  respect  to  the  X-axis. 

As  another  example,  consider  the  single-valued  function 

7/  =  5  .i*  —  6  —  or.  (1) 

The  following  is  a  table  of  corresponding  values : 
x=.      0,       1,  2,  I,  3,       4,       5,  .... 
y  =  -Q,   -2,  0,  i    0,   -2,   -6,  .... 

The  graph  is  shown  in  Fig.  27,  where  the  units  on  the  two  axes 
are  the  same.     We  now  see  that  the  curve  is  symmetrical  with 


Fig.  27 


38 


GRAPHIC   REPRESENTATION 


[Chap.  III. 


respect  to  a  line  parallel  to  the  F-axis  and  passing  through  the 
point  (f,  0). 

The  symmetry  is  also  shown,  without  plotting,  by  solving 
equation  (1)  for  x.     Thus, 

x  =  ^±  Vi-y,  (2) 

and  therefore,  for  a  given  value  of  7j,  x  has  two  values  represented 
by  points  equidistant  from,  and  on  opposite  sides  of,  the  point 

cc  =  f .  We  thus  see  that  the  line 
parallel  to  the  F-axis  through  the 
point  X  =  ^  bisects  the  chords  of 
the  curve  drawn  perpendicular  to  it. 

Notice  that  the  function  has  a 
turning  point  at  x  =  ^,  that  the 
value  of  y  is  there  equal  to  \,  and 
that  this  is  the  maxiraum  value  of  y. 

A  curve  is  symmetrical  tvith  respect 
to  a  point  if  the  point  bisects  all  the 
chords  of  the  curve  drawn  through  it. 

For  example,  consider  the  func- 
tion y  =  a^.  The  graph  is  shown 
in  Fig.  28,  where  the  unit  of  the 
scale  on  the  X-axis  is  taken  five 
times  as  great  as  the  unit  of  the 
scale  on  the  F-axis.  We  see  that 
the  origin  bisects  all  the  chords  of 
the  curve  drawn  through  it.  Hence 
the  curve  is  symmetrical  with  re- 
FiG.  28  spect  to  the  origin. 

30.  Intercepts.  The  distances  from  the  origin  to  the  points 
where  a  graph  crosses  the  axes  are  called  the  intercepts.  Thus, 
in  Fig.  27,  the  curve  crosses  the  X-axis  twice,  at  two  and  three 
units  to  the  right  of  the  origin.  The  X-intercepts  are  -}-  2  and 
4-  3.  The  curve  crosses  the  Y-axis  six  units  below  the  origin. 
The  y-intercept  is  —  6. 

The  X-intercepts  are  the  roots  of  the  equation  of  the  graph  when  y 
is  pxit  equal  to  zero,  and  the  Y-intercepts  are  the  roots  of  the  equation 
lohen  X  is  put  equal  to  zero. 


it 

^^l 

t    L 

V    ± 

t    X 

:     7 

'    f 

1P5-                       /-     ^ 

->-'                 '    ^3 

._ _..     _   .^     t-Jt- 

V      2    ' 

i  f--t 

4     ^        t 

t^t        ' 

>    2     -, 

'/       I 

Jt     ^?- 

_  _  _     Jj.  t  1- 

Tt    ^    J 

A^tie,* 

«v''<^' 

t           r.    ■Si''''' 

jyi^'^  y// 

^^"^^  ^It 

/      '/    J  J 

f  X         J 

if^      /    / 

4             t      ' 

t         i     i~            ~                                         --     - 

t         /-     f- 

t     J     t     - 

'      ^     -J 

-  J    t           : 

i/  4              ..  _       _ 

w     t 

'       4 

f 

t-j 

JZ  t 

1  / 

1  / 

/ 

y                             ± 

Art.  31]  GRAPH  IN  POLAR   COORDINATES  39 

EXERCISES 

1.  Draw  the  graph  of  the  function  y  =  x"^  -  2  x  —  S.  Find  the  position 
of  the  line  of  symmetry,  the  intercepts,  and  the  coordinates  of  the  turning 
point. 

2.  Draw  the  graph  of  the  function  y,  when  y^  —  2y  =  2x—l.  Find  the 
position  of  the  line  of  symmetry  and  the  intercepts.  Is  y  a.  single-valued,  or 
multiple-valued  function  of  a*  ? 

3.  Given  yx  =  4.  Show  that  the  line  bisecting  the  first  and  third  quad- 
rants is  a  hne  of  symmetry.  Find  the  coordinates  of  the  points  where  this 
line  meets  the  curve.  Is  y  a  monotone  function  of  a;  ?  A  single-valued  func- 
tion oi  X? 

4.  Show  that  the  graph  of  y  =(x—  1)'^  +  2  is  symmetrical  with  respect 
to  the  point  (1 ,  2) . 

5.  Show  that  if  an  equation  contains  only  even  powers  of  y,  the  graph  is 
symmetrical  with  respect  to  the  X-axis  ;  and  if  it  contains  only  even  powers 
of  X,  the  graph  is  symmetrical  with  respect  to  the  F-axis. 

6.  If  2/  =  ax"^  +  bx  +  c,  find  the  coordinates  of  the  turning  point. 

7.  A  rectangle  is  inscribed  in  a  circle  of  radius  5.  Express  the  area  of 
the  rectangle  as  a  function  of  the  length  of  one  side.  Draw  the  graph  of  the 
function  thus  found,  and  find  the  coordinates  of  the  turning  point.  What 
is  the  length  of  the  side  of  the  rectangle  of  maximum  area  inscribed  in  the 
circle  ? 

8.  A  box  is  to  be  constructed  having  a  square  base  and  containing  108  cubic 
feet.  The  box  is  to  have  no  cover.  Express  the  number  of  square  feet  of 
lumber  required  as  a  function  of  the  length  of  the  side  of  the  base.  Draw 
the  graph  of  the  function  obtained  and  locate  the  turning  point.  What  are 
the  coordinates  of  the  turning  point  ?  What  is  the  size  of  the  box  requiring 
the  least  amount  of  lumber  to  construct  it  ? 

31.  Graph  in  polar  coordinates.  Let  r  be  given  as  a  function 
of  0,  then  corresponding  values  of  the  independent  variable  and  of 
the  function  can  be  regarded  as  polar  coordinates  of  points.  When 
r  and  6  are  connected  by  an  equation,  a  table  of  corresponding 
values  can  be  computed  and  plotted  as  in  rectangular  coordinates. 
The  totality  of  radii  obtained  in  this  way  forms  a  graphical  repre- 
sentation of  the  function,  and  a  smooth  curve  drawn  through  the 
plotted  points  is  the  graph  of  the  function  in  polar  coordinates.  For 
example,  let  the  function  be  given  by  the  equation 


40 


GRAPHIC   REPRESENTATION 


[Chap.  III. 


The  following  is  a  table  of  corresponding  values,  6  being  measured 
in  radians  : 

/J  p,  TV  TT  TT  2i  IT  b  TV 

^"    '        6'  3'  2'  T'         "6"'         '''      ■■■• 

r  =  0,  1.0472,  2.0944,  3.1416,  4.1888,  5.2360,  6.2832,  .... 

A  part  of  the  graph  is  shown  in  Fig.  29.     The  curve  is  called  a 
spiral  of  Archimedes,  after  its  discoverer. 


Fig.  29 


EXERCISES 

1.  Draw  the  graph  of  the  equation  r  =  -  and  compare  with  the  graph  in 

the  preceding  article. 

2 

2.  Draw  the  graph  of  the  function  r  —  -■  The  curve  is  called  the  recip- 
rocal spiral. 

3.  How  does  the  graph  of  r  =  2  9  +  1  differ  from  the  graph  in  the  preced- 
ing article  ? 

4.  If  the  abscissa  of  every  point  in  Fig.  27,  Art.  29,  is  diminished  by  2\ 
units,  how  will  this  affect  the  graph  ?  How  will  it  affect  the  equation  ? 
Write  the  new  equation  and  draw  the  graph.  Compare  the  graph  with  those 
in  Arts.  27  and  28. 

5.  In  the  spiral  of  Archimedes,  let  the  radius  vector  rotate  in  the  negative 
direction.    Draw  the  curve  and  compare  with  the  graph  in  Art.  31. 


Arts.  32-34]      TRANSCENDENTAL  FUNCTIONS  41 

6.  Draw  the  graph  of  ?•  =  3  6.  How  does  this  curve  differ  from  the  spiral 
of  Archimedes  in  Art.  31  ? 

7.  In  Fig.  29,  the  curve  vflll  cross  the  initial  line  when  6  is  any  integral 
multiple  of  w.  Why  ?  What  is  the  distance  between  any  two  consecutive 
points  of  crossing  ? 

8.  Draw  the  graph  of  a;  =  a6,  where  a  is  any  constant  number.  For  what 
values  of  d  does  the  graph  cross  the  initial  line  ?  What  is  the  distance  be- 
tween any  two  consecutive  points  of  crossing  ? 

32.  Algebraic  functions.  If  the  function  and  the  independent 
variable  are  connected  by  an  algebraic  equation,  that  is,  an  equa- 
tion involving  only  a  finite  number  of  the  fundamental  operations 
of  addition,  subtraction,  multiplication,  division,  involution,  and 
evolution,  the  function  is  called  an  algebraic  function.  Thus,  for 
example,  in  each  of  the  equations  y=2  x-,  y-=2  x,  y"-—  oy  -\-  x=0, 
cc?  —  3  xy  4-2/^  =  0,  y  is  an  algebraic  function  of  x. 

To  find  the  value,  or  values,  of  an  algebraic  function  for  any 
given  value  of  the  independent  variable,  it  is  usually  necessary 
to  solve  an  algebraic  equation.  For  example,  if  ?/^  —  5  ?/  +  ^  =  0, 
it  is  necessary  to  solve  a  quadratic  equation  to  find  the  values  of 
y  for  any  given  value  of  x. 

33.  Transcendental  functions.  In  many  cases  of  great  practical 
importance,  the  function  and  the  independent  variable  are  not 
connected  by  an  algebraic  equation,  and  then  the  function  is 
called  a  transcendental  function.  The  simplest  examples  of  trans- 
cendental functions  are  furnished  by  the  trigonometric  functions 
and  logarithmic  functions.     Thus, 

y  =  sin  x  and  y  =  log  x 

are  transcendental  functions. 

To  find  the  value  of  a  transcendental  function  for  a  given  value 
of  the  independent  variable,  use  is  made  of  a  table.  We  thus 
have  tables  of  logarithms  and  tables  of  trigonometric  functions. 

34.  Graphs  of  transcendental  functions.  Corresponding  values 
of  function  and  independent  variable  can  be  taken  directly  from 
the  table  and  the  function  exhibited  graphically  in  rectangular 
coordinates  or  in  polar  coordinates,  as  in  the  preceding  articles. 


42 


GRAPHIC   REPRESENTATION 


[Chap.  III. 


EXERCISES 

1.    Draw  the  graphs  of  the  following  functions.     State  which  are  algebraic 
functions  and  which  are  transcendental  functions. 


(a)  y  =  tan  x,  (h)  y^  =  x'^, 

(d)  y^  =  4  a;2,  (e)  y  =  log  x, 

2.   Draw  the  polar  graphs  of  the  following  functions 


(ffl)  r  = 


sin  d 


(&)  r  =  2  a(l  —  cos  i 


(c)  y  =  cos  X, 
(/)  x^  +  y-  =  2x. 


(c)  r  =  (2(1  +  cos  ( 


3.  Using  the  relations  between  rectangular  and  polar  coordinates  (Art.  9), 
change  the  equations  in  exercise  2  to  rectangular  coordinates  and  plot  ?/  as  a 
function  of  x. 

4.  Change  the  equation  (/)  of  exercise  1  to  polar  coordinates  and  plot  r 
as  a  function  of  d. 


0 

^r 

^^•"^ 

^"^-s^ 

y^^ 

"^ 

y^ 

Jr" 

r 

1 

\ 

/ 

A\X 

\ 

/ 

( 

/ 

\ 

TT 

1                A 

D    IB 

0 

D'                         F\ 

' 

\^ 

_y 

w 

Fig.  30 

35.  Geometric  construction  of  the  graphs  of  trigonometric  func- 
tions. The  graphs  of  trigonometric  functions  can  be  constructed 
geometrically  without  the  use  of  tables.  For  example,  to  con- 
struct the  graph  oi  y  —  sin  cc,  let  0  be  the  origin  (Fig.  30)  and  F 
the  point  representing  -k  on  the  scale  OX.  With  any  point  A  on 
OX.  as  center  and  the  unit  of  the  scale  on  the  Y'-axis  as  radius, 
draw  a  circle.  Let  BAP  be  any  angle  x  measured  in  degrees. 
The  perpendicular  DP  is  then  sin  x.  Take  the  distance  OJy  so 
that 


OU  :  OF 


:180° 


then  the  point  U  represents  the  angle  x  measured  in  radians  on 
the  scale  OX.  Through  D'  draw  the  perpendicular  to  OX  and 
through  P  the  parallel  to  OX.     Let  these  lines  meet  in  P',  then 


Art.  35]  GEOMETRIC  CONSTRUCTION  OF  FUNCTIONS      43 

D'P'  =  DP=  sin  X.  Hence,  as  P  describes  the  circle,  P'  describes 
the  graph  oi  y  =  s\n  x. 

For  convenience,  divide  OF  into  a  number  of  equal  parts  and 
erect  perpendiculars  to  OX  through  the  points  of  division,  then 
divide  the  quadrant  BPC  into  the  same  number  of  equal  parts 
and  draw  parallels  to  OX  through  the  points  of  division.  Each 
perpendicular  meets  its  corresponding  parallel  in  a  point  of  the 
graph,  as  indicated  in  the  figure. 

The  graph  of  y  =  sin  x  is  called  the  sinusoid,  or  wave  curve. 

Trigonometric  functions  are  periodic  functions ;  that  is,  the 
value  of  the  function  is  repeated  again  and  again  for  values  of 
the  variable  which  differ  by  a  constant.  Thus  sin  x  has  the  same 
value  when  x  is  increased  or  decreased  by  any  integral  multiple  of 
2  TT.  Many  of  the  phenomena  in  nature  are  also  periodic.  For 
this  reason,  trigonometric  functions  are  of  great  importance  in 
the  applications  of  mathematics. 

EXERCISES 

1.  By  measuring  angles  from  the  line  CJ.,  Fig.  30,  instead  of  from  the 
line  BA^  show  how  to  construct  geometrically  the  graph  oi  y  =  cos  x. 

2.  In  Fig.  30,  draw  the  tangent  to  the  circle  at  B  and  let  it  meet  the 
radius  AP  produced  in  K.  Then  BK  is  tan  BAP  {AB  =  1);  show  how  to 
construct  geometrically  the  graph  oiy  =  tan  x. 

3.  Taking  CA  for  the  initial  line,  show  how  to  construct  the  graph  of 
y  =  cot  X. 

4.  Devise  a  method  for  constructing  geometrically  the  graphs  oi  y  =  sec  x 
and  y  =  cosec  x. 

5.  How  can  the  graph  of  a  trigonometric  function  be  used  to  find  the 
value  of  the  function  for  any  given  value  of  the  variable  ? 

6.  A  point  P  describes  a  circle  of  radius  a  with  the  uniform  velocity  of  k 
radians  per  second.     Show  that  the  period^  that  is,  the  time  of  one  complete 

9  -jT 

revolution,  is  T  =  - — 
A; 

7.  Let  the  center  of  the  circle  in  the  preceding  exercise  be  the  origin  of 

rectangular  coordinates.     Show  that,  at  the  end  of  t  seconds,  the  coordinates 

of  the  point  P  are  ^ 

X  =  a  cos  kt  =  a  cos  - — . 
T 

y  =  a  sm  M  =  a  sin 


44  GRAPHIC   REPRESENTATION  [Chap.  III. 

The  kind  of  motion  described  by  eitlier  of  these  equations  is  called  a 
simple  harmonic  motion  (S.  H.  M.),  a  is  called  the  amplitude  and  T  the 
period  of  the  S.  H.  M. 

36.  The  exponential  function.  When  the  function  and  the  in- 
dependent variable  are  connected  by  the  equation 

y  is  called  an  exponential  function  of  x.  The  constant  a  is  called 
the  base.  The  exponential  function  is  transcendental,  since  y  and 
X  are  not  connected  by  an  algebraic  equation. 

37.  Graph  of  the  exponential  function.  The  graph  of  the  ex- 
ponential function  can  be  constructed  as  follows :  From  a  point 
A  on  the  X-axis  (Fig.  31)  lay  off  a  unit  AB  and  erect  the  ordinate 
BBi  equal  in  length  to  the  base  a.  Draw  the  line  AB^  and  also 
the  line  AZ  making  an  angle  of  45°  with  the  X-axis.  Through 
jBj  draw  the  parallel  to  the  X-axis  meeting  AZ  in  C2,  and  through 
O2  the  perpendicular  to  the  X-axis  meeting  AB^  in  C^  and  the 
X-axis  in  C.  The  segment  CCi  is  equal  in  length  to  a^;  that  is, 
the  value  of  the  function  when  x  is  2.      For,  by  similar  triangles, 

AB:BB,::AC:CO„ov 
1:  a:  :  a:  CCi, 

since  AC=  CC^  =  BB^^  =  a.     Hence,  CC^  =  al 

Similarly,  drawing  the  parallel  to  the  X-axis  through  Ci  and 
the  perpendicular  through  C3,  we  can  prove  that  DD^  is  equal  in 
length  to  al  Thus  all  the  positive  integral  powers  of  a  can  be 
constructed  geometrically.  The  negative  integral  powers  can  also 
be  constructed  by  means  of  the  parallels  and  perpendiculars. 
Thus,  JOfi  =  cr'^,  KK^  =  cr"^,  etc. 

Let  0  be  the  origin  of  coordinates  and  OT  the  F-axis.  Con- 
struct parallels  to  the  F-axis  at  intervals  of  a  unit,  thus  forming 
a  series  of  rectangles  with  the  parallels  to  the  X-axis.  The 
graph  oi  y=  a"  cuts  through  opposite  corners  of  these  rectangles, 
beginning  from  the  point  (0,  1)  and  running  each  way. 

The  exponential  function  has  no  turning  points  and  is  therefore 
a  monotone  function  (Art.  27).  It  is  important  in  representing 
physical  phenomena  which  are  not  periodic,  such,  for  example,  as 


Arts.  37,  38] 


INVERSE   FUNCTIONS 


45 


the  retarding  effect  of  friction,  the  pressure  of  the  atmosphere  as 
a  function  of  the  altitude,  etc.     The  exponential  function  is  also 


E 

■ 

/ 

z 

D. 

/ 

/ 

C, 

c, 

/ 

2?. 

/ 

/ 

^3 

a' 

a' 

il/i 

/ 

C\ 

a'- 

/ 

^,^-^ 

d 

K,£ 

,-^1 

^m 

-|«-ia-l 

^ 

A  K. 

^i 

B  C 

7      J 

0         1 

7                -2      -1 

0 

I        \ 

- 

X 

Fig.  31 

important  in  computing  interest  tables,  since  the  amount  y,  of  $  1 
for  X  years,  at  rate  i  compound  interest,  is  given  by  the  formula 

y  =  (l  +  iy. 

Frequently  the  base  is  taken  to  be  e  =  2.71828  •••.  The  num- 
ber e  is  the  base  of  the  natural,  or  naperian,  system  of  log- 
arithms. 

38.  Inverse  functions.  If  two  variables  are  connected  by  an 
equation,  or  otherwise,  either  variable  may  be  regarded  as  a 
function  of  the  other.  For  example,  in  the  equation  y  =  2  x^,  we 
think  of  y  as  the  function  and  x  as  the  independent  variable,  but 
we  may  regard  x  as  the  function  and  y  as  the  independent  variable. 
Either  of  these  functions,  y  or  x,  is  called  the  inverse  of  the 
other. 

It  is  convenient  to  retain  the  notation  "  y  means  function  and  x 
means  independent  variable."  Hence,  to  obtain  the  equation  de- 
fining the  inverse  of  a  given  function  y,  we  have  but  to  inter- 
change X  and  y  in  the  given  equation  and  then  express  y  in  terms 


46  GRAPHIC   REPRESENTATION  [Chap.  III. 

of  X.     Thus,  in  the  above  example,  the  inverse  function  is  defined 
by  the  equation  ,- 

x  =  2y\ovy  =  ±  ^| 

Similarly,  the  equations 

y  =  sin  X  and  x  =  sin  y,  or  y  =  arc  sin  x, 

define  a  pair  of  inverse  functions.     Again,  the  equations 

y  =  a"  and  x  =  a^,  or  y  =  log^  x, 
define  a  pair  of  inverse  functions. 

EXERCISES 

1.  Draw  the  graph  representing  the  amount  of  $1  at  [>%  compound  in- 
terest as  a  function  of  the  time,  interest  being  compounded  annually. 

2.  Show    that     the    following    pairs    of     equations     represent    inverse 
functions  : 

(a)  2/  =  3  x^  and  y  =  'Y o'        (6)  y  =  5x  —  6  —  x^  and  ?/  =  |  ±  Vi  -  x, 

(c)  y  =  a^^  and  y  =     -"''    ,       (cl)  y  =  tan  2  x  and  y  =  ^  arc  tan  x. 
logft  a 

3.  Write  the  inverse  of  each  of  the  following  functions. 


(a)  2/  =  cos 3. T,  (6)?/  =  -^tan| 

(c)   y  =r  loge  -,  (d)  2/  =  x2  -  5  iC  +  6. 

4.  Show  how  to  construct  the  graph  of  y  =  ffl"^  from  the  graph  oi  y  =  a^ 
in  Art.  37.     How  will  changing  the  sign  of  x  affect  any  graph  ? 

5.  Given  the  graphs  oi  y  =  a^  and  y  =  a~^  on  the  same  coordinate  axes, 
how  can  one  construct  geometrically  the  graph  oi  y  =  ^   ~*"  ^ —  ? 

6.  With  the  graphs  oi  y  =  sin  x  and  y  =  cos  x  on  the  same  coordinate 
axes,  construct  geometrically  the  graph  of  ?/  =  sin  x  +  cos  x. 

39.  Graph  of  an  inverse  function.  Since  the  inverse  of  a  given 
function  is  obtained  by  interchanging  x  and  y,  the  graph  of  the 
inverse  function  can  be  constructed  by  interchanging  the  coordi- 
nates of  every  point  on  the  graph  of  the  given  function.  Thus, 
if  P  (Fig.  32)  is  a  point  on  the  graph  of  the  given  function,  P'  is 
a  point  on  the  graph  of  the  inverse  function  when 

OD'  =  DP  and  D'P'  =  OD. 


Art.  39]         GRAPH  OF  AN   INVERSE   FUNCTION 


47 


By  this  construction,  P  and  P'  are  symmetrically  situated  with 
respect  to  the  line  OA  bisecting  the  first  and  the  third  quadrants. 
As  P  describes  the 
graph  of  the  given 
function,  P'  de' 
scribes  the  graph  of 
the  inverse  function. 
Hence,  having  given 
the  graph  of  any 
function,  we  can  ob- 
tain the  graph  of  the 
inverse  function  by 
plotting  points  sym- 
metrically situated 
to  the  points  of  the 
given  graph  with  re- 
spect-to the  line  OA. 
Or  we  may  consider 
the  entire  jDlane  ro- 
tated through  180°  about  the  line  OA,  carrying  the  given  graph 
with  it.  The  new  position  of  the  graph  is  the  graph  of  the  in- 
verse function.  Por  example,  let  JSLX  be  the  graph  of  y  =a^ 
(Fig.  32),  where  a  is  taken  to  be  2.  Eotating  the  plane  about  OA, 
the  curve  assumes  the  position  RS,  symmetrical  to  MN  with  re- 
spect to  the  line  OA.  Therefore  RS  is  the  graph  of  the  inverse 
function  y  =  log^  x. 


Fig.  32 


EXERCISES 

1.  Given  y  =  5  x  —  6  —  x-,  draw  the  graph  of  the  inverse  function. 

2.  Construct  the  graphs  of  the  following  functions  : 

(a)  y  =  arc  sin  x,         (6)    y  =  arc  tan  x,         (c)    y  =  arc  cos  x, 

2  3 

(fZ)    y  =  x^  and  y  =  x^. 

3.  Show  that  the  graph  oi  y  —  -  and  the  graph  of  the  inverse  function 

X 

coincide  throughout.     What   condition  must  be  satisfied  in  order  that  the 
graph  of  any  function  shall  coincide  with  the  graph  of  its  inverse  ? 


48 


GRAPHIC   REPRESENTATION 


[Chap.  Ill, 


40.  Observation.  A  functional  correspondence  between  two  variables 
is  often  established  by  observation  when  no  relation  between  the  variables  is 
known.  Thus  the  temperature  at  a  given  place  can  be  observed  throughout 
the  day  and  the  results  tabulated.  The  temperature  can  then  be  regarded 
as  a  function  of  the  time,  and  the  functional  relationship  can  be  graphically 
exhibited  as  in  the  preceding  articles.  In  such  cases  the  functional  relation- 
ship is  given  in  the  form  of  a  table  of  corresponding  values. 

41.  Machines.  Machines  are  devised  to  draw  the  graph  automatically 
and  thus  avoid  the  necessity  of  making  repeated  observations.  For  example, 
tlie  weather  bureau  has  an  instrument  to  graph  the  temperature  as  a  func- 
tion of  the  time.  Coordinate  paper  is  wound  upon  a  clock-driven  drum,  and 
a  pen  is  connected  with  a  thermometer  in  such  a  way  that  the  rise  and  fall 
of  temperature  is  recorded  upon  the  paper  at  the  proper  time.  The  record 
exhibits  the  functional  relationship  in  the  form  of  a  graph.  Corresponding 
values  of  the  function  and  the  independent  variable  can  be  read  from  the 
graph  as  readily  as  from  a  table. 

Other  records  exhibit  functional  relationship  in  the  form  of  a  graph  upon 
polar  coordinate  paper. 

EXERCISES 

1.  The  following  table  shows  the  length  of  a  rubber  cord  in  centimeters 
when  stretched  by  a  weight  in  kilograms  attached  to  one  end.  Draw  a  curve 
representing  approximately  the  graph  of   the  length  as  a   function   of   the 

weight. 

Weight  ....  0 

Length  ....  10 

Weight  ....  3.5 

Length  ....  12.2 

2.  The  number  of  deaths  per  hundred  thousand  lives,  according  to  the 
American  experience  table  of  mortality,  is  as  follows  : 


.5 

1.0 

1.5 

2.0 

2.5 

3.0 

10.1 

10.3 

10.6 

10.9 

11.3 

11.7 

4.0 

4.5 

5.0 

5.5 

6.0 

12.7 

13.3 

13.9 

14.6 

15.3 

Age 

Number  of  Deaths 

Age 

NuMBEE  OF  Deaths 

20 

781 

60 

2669 

25 

807 

65 

4013 

30 

843 

70 

6199 

35 

895 

75 

9437 

40 

■  979 

80 

14447 

45 

1116 

85 

23555 

50 

1378 

90 

45455 

55 

1857 

95 

100000 

Art.  41] 


MACHINES 


49 


Draw  a  curve  representing  the  graph  of  the  number  of  deaths  as  a  func- 
tion of  the  age. 

3.   The  net  annual  premium  for  an  assurance  of  §  1000  for  life,  according 
to  the  American  experience  table  of  mortality,  interest  at  3  %,  is  as  follows  : 


Age 

Premium 

Age 

Pkemu  M 

20 

$  14.41 

40 

.$24.75 

25 

^16.11 

45 

$29.67 

30 

$18.28 

50 

$36.36 

35 

f  21.08 

Draw  a  curve  representing  the  graph  of  the  premium  as  a  function  of  the 
age. 

4.    The  cost  of  a  money  order  depends  upon  the  amount  as  follows  : 


Amount 

Cost 

Amount 

Cost 

$0       to  .'$2. 50 

3ct. 

$  30  to  $  40 

15  ct. 

$2.60  to  $5 

5ct. 

$  40  to  $  50 

18  Ct. 

$  5       to  $  10 

8  ct. 

$  50  to  $  60 

20  Ct. 

$  10     to  $  20 

10  ct. 

$  60  to  $  75 

25  ct. 

$20     to  $30 

12  ct. 

$  75  to  $  100 

30  ct. 

Draw  a  curve  representing  the  graph  of  the  cost  as  a  function  of  the 
amount. 

5.  Figure  33  A  (p.  50)  represents  a  thermograph  for  April  12,  13, 
and  14.  The  ordinates  are  made  curvilinear  to  allow  for  the  pivotal  motion 
of  the  drawing  pen. 

Determine  the  maxima  and  minima  temperatures  between  noon  of 
April  12  and  noon  of  April  14.  When  was  the  temperature  highest  ?  When 
lowest  ? 

6.  Figure  33  B  (p.  51)  is  a  steam  pressure  gauge  on  polar  coordinate 
paper.  The  radii  are  made  curvilinear  to  allow  for  the  pivotal  motion  of 
the  drawing  pen. 

Determine  the  time  of  greatest  pressure.     The  time  of  least  pressure. 

7.  By  means  of  the  table  of  exponential  functions  (page  V),  make  a 
careful  drawing  of  the  graph  of  y  =  e^.  From  the  graph  thus  made  con- 
struct the  graph  ot  y  =  e~^.  Compare  the  readings  from  the  graph  with  the 
values  of  e~^  taken  from  the  table. 

From  the  graph  of  y  =  e^,  construct  the  graph  of  y  =  —  e''. 


50 


GRAPHIC   REPRESENTATION 


[Chap.  Ill- 


8.  Having  given  the  grapli  of  y  =  f(x)^  show  how  to  obtain  the  graphs 
oiy=f{-x),  y  =-f(x),  and  y  =-f(-x). 

9.  According  to  Boyle's  law,  tlie  volume  of  a  gas  is  inversely  propor- 
tional to  the  pressure  which  it  sustains.  If  a  volume  of  4  cubic  feet  sustains  a 
pressure  of  1  atmosphere,  write  the  equation  expressing  the  volume  as  a 
function  of  the  pressure.     Draw  the  graph  of  this  function. 

10.    The  increase  in  length  of  a  metal  bar  is  proportional  to  the  tempera- 
ture to  which  the  bar  is  subjected.     If  the  bar  is  1  foot  long  at  0°  temperature 


Fig.  33  A 


and  1.0004  feet  long  at  20°  temperature,  write  the  equation  expressing  the 
length  as  a  function  of  the  temperature.     Draw  the  graph  of  this  function. 

11.  The  intensity  of  light  is  inversely  proportional  to  the  square  of  the 
distance  from  the  source  of  the  light.  Write  the  equation  expressing  the 
intensity  as  a  function  of  the  distance.  Draw  the  graph  of  this  function. 
If  the  intensity  of  light  at  a  point  on  the  earth  directly  underneath  the  sun 
is  taken  as  the  unit  of  intensity,  calculate  the  intensity  of  light  on  the  planet 
Venus  at  a  point  directly  underneath  the  sun.  Take  the  distance  from  the 
earth  to  the  sun  as  93,000,000  miles  and  the  distance  of  Venus  from  the  sun 
as  67,000,000  miles. 


Art.  41] 


MACHINES 


51 


Fig.  33  B 

12.   The  water  rates  of  a  certain  city  depend  upon  the  amount  consumed 
and  are  as  follows  : 


Consumption  per  Day 

Rate  pek  1000  Gallons 

0  to    499  gallons 

.35 

500  to    999  irallons 

.32 

1000  to  1999  gallons 

.28 

2000  to  2999  gallons 

.24 

3000  to  3999  gallons 

.20 

4000  to  4999  gallons 

.15 

5000  to  5999  gallons 

.12 

6000  and  over  gallons 

.10 

52  GRAPHIC   REPRESENTATION  [Chap.  III. 

Calculate  the  monthly  (30  clays)  bill  of  a  consumer  and  draw  a  curve 
representing  the  graph  of  the  amount  of  the  bill  as  a  function  of  the  number 
of  gallons  consumed  per  month. 

13.  A  certain  mixture  of  concrete  contains  1.4  barrels  of  cement  per 
cubic  yard  of  concrete.  If  the  cement  costs  $  1.20  per  barrel  and  the  sand 
and  crushed  stone  costs  82.10  per  cubic  yard,  write  an  equation  expressing 
the  cost  of  the  concrete  as  a  function  of  the  number  of  cubic  yards.  Draw 
the  graph  of  this  function. 

14.  Express  the  area  of  a  circle  as  a  function  of  the  radius  and  draw  the 
graph  of  the  function. 

15.  Draw  the  graphs  of  y  =  sin  a;  and  y  =  cosx  on  the  same  coordinate 
axes.    From  these  graphs  construct  the  graph  of  the  function 

y  =  2smx  +  cos  x. 


CHAPTER   IV 
LOCI  AND   THEIR  EQUATIONS 

42.  Locus  of  a  point,  equation  of  locus.  When  a  point  P{x,  y) 
moves  in  the  plane,  the  path  it  describes  is  called  the  locus  of 
the  point.     The  coordinates  x,  y  are  then  variables  (Art.  22). 

If  the  point  P{x,  y)  moves  according  to  a  given  law,  this  law 
will  lead  to  an  equation  connecting  x  and  y  called  the  equation  of 
the  locus.  The  equation  of  the  locus  defines  ?/  as  a  function  of  x, 
and  the  locus  itself  is  the  graph  of  this  function.  As  an  example, 
suppose  P  moves  so  that  it  is  constantly  at  a  fixed  distance  from 
a  fixed  point  A.  We  know,  then,  that  the  point  describes  a  circle. 
This  circle  is  the  locus  of  the  point  P.  The  point  P  moves 
according  to  the  given  law 

AP  =  constant, 

and  we  shall  see  that  this  law  leads  to  an  equation  connecting 
the  variable  coordinates  x  and  y. 

43.  A  fundamental  problem.  When  the  law  which  governs  the 
motion  of  a  point  is  given,  a  fundamental  problem  presents  itself  • 
namely,  to  find  the  equation  of  the  locus.  For  example,  suppose 
a  point  moves  so  that  it  is  always  equidistant  from  the  points 
F  =  (l,  2)  and  F,={3,  1)  (Fig.  34).  To  find  the  equation  of  the 
locus,  let  P{x,  y)  be  any  point  equidistant  from  F  and  F-^.  Then, 
by  the  given  law,  ^^  ^  ^^^^ 

for  all  positions  of  P.     But 

PF=-V{x  -  iy-\-{y  -  2)2  and  PF,  =  V(x  -  3y+  {y  -  If. 
Therefore  we  have 


V(a.-  -  1)2+  (,y  -  2f=^ix  -  3Y+{y  -  If,  (2) 

53 


54 


LOCI  AND   THEIR  EQUATIONS  [Chap.  IV. 


which  reduces  to  4:X  —  2y  —  5  =  0.  (3) 

This  is  the  required  equation.     The  locus  is  the  perpendicular 
bisector  of  the  segment  FF^. 

The  following  property  and  its  converse  are  characteristic  of  this 
locus  and  its  equation ;  namely,  the  coordinates  of  every  point 
on  the  locus  satisfy  equation  (3).     For,  if  the  point  is  on  the 


IJ-                ' 

\           Ml 

-It     -          ^ 

1            M 

1      '      ;   1             1    ' 

1      III 

/^      1  1  i 

1       /  1  ■       !    !    ' 

/      1         :     1         ] 

1 

1 

,j  JlCxilti 

11 

1                '         • 

n 

1                II 

^■^  1 

7 

y       . 

y^         i   / 

<                 / 

-^  -        -                       : 

J'           SsJ       / 

1*  C   - 

-    ^ 

55>  X 

:              TT 

^r 

7 

^^ 

"                    "Z 

. ...__!rt- 

-           U-                        t 

A. 

t 

t       -- 

t 

~     V          " 

._    :                :  t      ..  _ 

A 

t 

i 

t' 

t 

1 

t 

f 

^ 

Fig.  34 


locus,  it  is  equidistant  from  F  and  F^.     Therefore  its  coordinates 
satisfy  (2)  and  consequently  (3). 

Conversely,  if  the  coordinates  of  any  point  satisfy  equation  (3), 
the  point  is  on  the  locus.  For  then  the  coordinates  of  the  point 
also  satisfy  equation  (2),  and  the  point  is  therefore  equidistant 
from  F  and  F^ ;  that  is,  the  point  is  on  the  locus. 

44.  General  definition.  The  property  just  proved  for  the 
special  locus  in  Fig.  34  leads  to  the  following  definition:  The 
equation  of  the  locus  of  a  point  is  an  equation  in  the  variables  x  and 
y  which  is  satisfied  by  the  coordinates  of  every  point  on  the  locus ; 


Arts.  44,  45] 


THE   CIRCLE 


55 


and  conversely,  every  point  whose  coordinates  satisfy  the  equation 
lies  on  the  locus. 

The  locus  of  a  point  moving  according  to  a  given  law  is,  in 
general,  a  curve,  and  we  shall  often  speak  of  the  equation  of  the 
locus  as  the  equation  of  the  curve.* 

The  problem  to  find  the  equation  of  the  locus  wlien  the  law 
governing  the  motion  of  the  point  is  given  will  be  illustrated  in 
the  succeeding  articles,  where  the  equations  of  a  number  of  im- 
portant curves  are  found  and  methods  given  for  constructing  the 
curves. 

45.  The  circle.  A  x>oint  moves  so  that  it  is  alivays  r  units  from 
ajixecl  jioint  (a,  6).     Find  the  equation  of  the  locus. 

The  locus  is  evidently  a  circle 
whose  radius  is  r  and  whose 
center  is  the  point  G  =  (a,  h) 
(Fig.  35). 

To  find  the  equation  of  the 
locus,  assume  that  P{x,  y)  is 
any  jDoint  r  units  from  C.  The 
point  P  is  then  on  the  locus. 
The  statement  of  the  law  gov- 
erning the  motion  of  P  is  then 

CP=r,  (1) 

for  all  positions  of  P.     But 


P{^,y) 


Fig.  35 


and  therefore 


GP  =  ^(-x-af  +  iy-by,  (2) 

(ic-a)2+ (I/- 6)2  =  ^2.  (3) 

If  the  center  of  the  circle  is  taken  at  the  origin  of  coordinates. 


then  a  =  0  and  b  =  0,  and  equation  (3)  becomes 

a52  +  ?/2  =  /'2. 


(4) 


Equations  (3)  and  (4)  are  standard  forms  of  the  equation  of  a 
circle.  The  student  should  test  equation  (3)  by  the  definition 
given  in  Art.  44. 

*The  word  "curve"  will  henceforth  be  used  to  denote  any  continuous  line, 
straiffht  or  curved. 


56  LOCI   AND   THEIR   EQUATIONS  [Chap.  IV. 

EXERCISES 

1.  Find  the  equations  of  tlie  following  circles  : 

(a)  Center  (0,  1)  and  radius  3.     (6)  Center  (—  2,  0)  and  radius  2. 
(c)  Center  (—  4,  3)  and  radius  3.     (cl)  Center  (1,  2)  and  radius  6. 

2.  Find  the  equation  of  the  circle  whose  center  is  (2,  3)  and  which  passes 
through  the  origin. 

3.  What  is  the  equation  of  the  circle  which  has  the  line  joining  the  points 
(3,  2)  and  (—  7,  4)  for  a  diameter  ? 

4.  Find  the  equation  of  the  circle  which  passes  through  the  three  points 
(0,  1),  (5,  1),  and  (2,-3). 

5.  A  point  moves  so  as  to  be  equidistant  from  the  points  (3,  —  1)  and 
(—  2,  3).     Draw  the  locus  and  find  its  equation. 

6.  Find  the  equation  of  the  perpendicular  bisector  of  the  segment  joining 
(a,  6)  to  (c,  cl). 

7.  A  point  moves  so  that  the  ratio  of  its  distances  from  the  points  (8,  0) 
and  (2,  0)  is  constantly  equal  to  2.     Find  the  equation  of  the  locus. 

8.  A  point  moves  so  that  the  sum  of  the  squares  of  its  distances  from 
(3,  0)  and  (—  3,  0)  is  constantly  equal  to  68.     Find  the  equation  of  the  locus. 

9.  A  circle  circumscribes  the  triangle  (6,  2),  (7,  1),  (8,  -  2).  Draw  the 
figure  and  find  the  equation  of  the  circle. 

46.  The  equation  x^  +  j/^  +  Asc  +  By  +  C  =  0  When  equation 
(3)  of  the  preceding  article  is  expanded  and  arranged  according 
to  the  powers  of  x  and  y,  it  takes  the  form 

x^  +  'if  +  Ax  +  By  +  C  =  0,  (1) 

where  A,  B,  and  C  are  constants  depending  upon  the  radius  of 
the  circle  and  the  coordinates  of  tlie  center. 

A  second  problem  now  arises :  Is  equation  (1)  the  equation  of 
a  circle  for  all  possible  values  of  A,  B,  and  C?  To  answer  this 
question,  we  shall  complete  the  squares  of  the  terms  in  x  and  y 
separately,  and  thus  put  equation  (1)  in  the  form 

In  this  form,  the  equation  states  that  the  length  of  the  segment 
joining  [ , )  to  {x,  y)  is  constantly  equal  to 


Arts.  46,  47]  THE   STRAIGHT   LINE  57 

for  all  positions  of  the  point  (x,  y).  Let  D  stand  for  the  ex- 
pression under  the  radical  in  (3) ;  then  we  can  draw  the  following 
conclusions : 

1.  If  Z)  >  0,  (1)  is  the  equation  of  a  circle  whose  center  is  the 

■   ,  f     A        B\  .      . 

point  (  — -,  — -^  ]  and  whose  radius  is  VZ). 

2.  If  Z)  =  0,  (1)  is  satisfied  by  the  coordinates  of  a  single 
point ;  namely,  the  point  (  —  — , )•     In  this  case  the  locus  is 

called  a  null  circle. 

3.  If  Z)  <  0,  there  is  no  point  in  tha  plane  whose  coordinates 
satisfy  (2)  and  consequently  no  point  whose  coordinates  satisfy 
(1).     In  this  case  the  locus  is  called  an  imaginary  circle. 

We  shall  find  it  convenient  to  say  that,  in  any  case,  equation 
(1)  is  the  equation  of  a  circle,  but  that,  in  particular  cases,  this 
circle  may  be  a  null  circle,  or  an  imaginary  circle. 

EXERCISES 

1.  Find  the  coordinates  of  the  center  and  the  radius  of  the  following 
circles.     Construct  the  figure  when  possible. 

{a)  x^  +  2/2-6  X  -  16  =  0.  (5)  ^2  +  y2  _  6  x  +  4  y  -  5  =  0. 

(c)  3x2  +  32/2-10x-242/  =  0.  {d)   (x  +  l)2  +  (?/ -  2)^=  0. 

(e)  x2  +  2/2  =  8  X.  (/)  7  x2  +  7  ?/2  -  4  x  -  ?/  =  3. 

ig)  «2  +  2/2  _  2  X  +  2  2/  +  5  =  0.  {h)  x~  +  2/2  +  16  X  +  100  =  0. 

2.  Eind  the  coordinates  of  the  center  and  the  radius  of  the  circle  which 
passes  through  the  points  (5,  —  3)  and  (0,  6)  and  has  its  center  on  the  line 
2x-32/-6  =  0. 

3.  A  point  moves  so  that  the  sum  of  the  scjuares  of  its  distances  from  two 
fixed  points  is  constant.     Prove  that  the  locus  is  a  circle. 

4.  A  point  moves  so  that  the  ratio  of  its  distances  from  two  fixed  points 
is  constant.  Prove  that  the  locus  is  a  circle  if  the  constant  ratio  is  different 
from  unity,  and  a  straight  line  if  the  constant  ratio  is  equal  to  unity. 

47.  The  straight  line.  A  point  moves  on  the  straight  line  joining 
the  fixed  points  P^  =(x-^,  y^  and  P,  =  (^'2;  y-i)-  Find  the  equation  of 
the  locus. 

Choose  any  point  P(x,  y)  on  the  straight  line  joining  P^  to  Pj 
(Fig.  36).     Then, 


58 


LOCI   AND   THEIR  EQUATIONS  [Chap.  IV. 


slope  of  segment  1\P  =  slope  of  segment  P1P2. 
But  (Art.  11), 

slope  of  FyP  =  -^^^ ,  and  slope  of  P.P^  =  ^  ~  ^^ 

Therefore 

2/ -  2/1  _  1/2 -2/1 


(1) 


iC  —  if  1      iCg  —  a?i 

The  expression  ^- ~  ■^^  is  called  the  slope  of  the  line.     Eepresent- 

1' m'. 


ing  the  slope  of  the  line  by  m,  equation  (1)  becomes 


(2) 


If  the  point  P^  =(0,  6),  that  is,  the  point  of  intersection  of  the 
line  with  the  F-axis,  equation  (2)  assumes  the  form 

y  =  mx  +  6.  (3) 

If  Pi  =  (0,  b)  and  Po={a,  0), 

then  m= ,  and  equation  (2) 

a 


reduces  to 


^2(3:2, y  2) 


a     b 


(4) 


^x 


Fig.  36 


Equations  (1),  (2),  (3),  and 
(4)  are  all  standard  forms  of 
the  equation  of  a  straight  line. 
Equation  (1)  is  called  the  two- 
point  form,  equation  (2)  is  the 
slope-point  form,  equation  (3)  is  the  slope  form,  and  equation  (4)  is 
the  intercept  form. 

From  these  equations,  we  conclude  that  the  equation  of  a  straight 
line  is  of  first  degree  in  the  variables  x  and  y. 

Conversely,  it  may  be  shown  that  any  equation  of  the  first  de- 
gree in  the  variables  x  and  y  is  the  equation  of  a  straight  line. 

Eor,  let  .  -r,       .     ^        rv  /ex 

'  Ax  +  By  +  C=Q  (5) 

be  such  an  equation.     Solving  for  y,  we  have 

A        C 

y  = X . 

^  B        B 


=  0  (1) 


Arts.  47,  48]  THE   DETERMINANT  FORM  59 

Comparison  with  equation  (3)  shows  that  (5)  must  be  the  equa- 

4 

tion  of  a  straight  line  whose  slope  is  —  —  and  whose  intercept  on 

C  ^ 

the   F-axis  is This  reasoning   fails  when  B  is  zero.     In 

IB 
that  case,  however,  equation  (5)  reduces  to  Ax  -\-  C  =0,  which  is 
the  equation  of  a  straight  line  parallel  to  the  F-axis,  since  x  has 

C 

the  constant  value for  all  values  of  y.     Hence,  in  every  case 

(5)  is  the  equation  of  a  straight  line. 

48.  The  determinant  form.     The  equation  of  a  straight  line  can 
be  written  in  the  form  of  a  determinant.     Thus,  the  equation 

X  y  1 
'Vi  yx  1 
X2     2/2     1 

is  the  equation  of  the  straight  line  joining  the  points  Pi  =  (a,'i,  y^ 
and  P2  =  (x2,  yo).  For,  equation  (1)  is  of  the  first  degree  in  x  and 
y  and  therefore  is  the  equation  of  some  straight  line,  by  the  pre- 
ceding article.  Moreover,  the  equation  states  that  the  area  of 
the  triangle  whose  vertices  are  [x,  y),  (x^,  y{),  and  (x.^,  yo)  is  zero 
(Art.  20).     Hence,  the  point  P(x,  y)  is  on  the  line  joining  P^  and 

EXERCISES 

1.  Write  the  equations  of  the  lines  passing  through  the  following  pairs  of 
points : 

(«)   (0,  1)    and    (5,6);  (6)   (1,-2)    and    (-3,4);    (c)   (5,   -2)    and 
(_  4,  —  1)  ;  (d)   (—  1,  3)  and  (3,  —  4).     Draw  the  figure  in  each  case. 

2.  Find  the  intercepts  which  each  of  the  lines  in  exercise  1  makes  upon 
the  coordinate  axes.     Write  the  equations  in  intercept  form. 

3.  With  the  intercept  on  the  F-axis  and  the  slope,  write  the  equation  of 
each  line  in  exercise  1  in  the  slope  form . 

4.  Write  the  equation  of  each  of  the  lines  in  exercise  1  in  the  determinant 
form. 

5.  Find  the  slope  and  the  intercepts  of  each  of  the  following  lines  : 


(a)    2ij  +  BX-1  =  x  +  2.-:^  (b) 


r  —  1      x-3 


2  3 


(e)    y^  =  S.  (d)  aLzil=2x_pi 


60 


LOCI  AND   THEIR   EQUATIONS  [Chap.  IV. 


6.  Write  the  equation  of  each  of  the  lines  in  exercise  5  in  the  intercept 
form.     In  the  slope  form. 

49.  The  ellipse.  A  point  moves  so  that  the  sum  of  its  distances 
from  tivo  fixed  points  F  and  F^  is  constantly  equal  to  2  a.  Construct 
the  locus  and  find  its  equation. 

In  the  first  place,  2  a  must  be  greater  than  the  length  of  the 
segment  FF^,  otherwise  no  locus  is  possible.     Lay  off  a  line  AiB^, 


^Y 


M       c 


Fig.  37 


2  a  units  in  length  (Fig.  37).  Take  C,  any  point  on  A^B^,  and 
with  A^C  as  radius  describe  a  circle  about  F.  With  CB^  as  radius 
describe  a  circle  about  F^.  The  two  circles  meet  in  the  points  3f 
and  iV.  These  points  are  on  the  locus,  since  the  sum  of  the  radii  of 
the  two  circles  is  2  a.  Taking  the  smaller  circle  about  F^  and  the 
larger  about  F,  two  more  points,  M'  and  JSf',  are  found  on  the 
locus.  By  taking  C  at.  different  places  on  AiB^,  as  many  points 
of  the  locus  can  be  found  as  may  be  desired. 

Another  construction  of  the  locus  is  made  as  follows :  stick 
pins  in  the  paper  at  the  points  F  and  F^.  Tie  the  ends  of  a 
string  together  so  that  the   loop  is  just  equal  to  2  a  plus  the 


Art.  49]  THE   ELLIPSE  61 

distance  from  F  to  Fy.  Drop  the  loop  over  the  pins  and  stretch 
it  taut  with  a  pencil  point.  Keeping  the  string  stretched,  move 
the  pencil  around  ;  it  will  describe  the  locus. 

This  locus  is  called  an  ellipse.  The  fixed  points  F  and  F^  are 
called  the  foci  of  the  ellipse.  The  distances  from  any  point  on 
the  ellipse  to  the  foci  are  called  the  focal  radii  of  the  point. 

To  find  the  equation  of  the  ellipse,  let  the  line  joining  the  foci 
be  the  X-axis,  and  the  perpendicular  bisector  of  the  segment  FF^, 
the  T^axis.  Let  P{x,  y)  be  any  point  on  the  ellipse  ;  then  PF=:  r 
and  PFi  =  Vi  are  the  focal  radii  of  P.     By  definition  we  have 

r  +  r,  =  2a  (1) 

for  every  position  of  P. 

Let  2  c  denote  the  length  of  the  segment  FF^ ;  then  the  coordi- 
nates of  F  and  Fj^  are  ( —  c,  0)  and  (c,  0),  respectively.     Then 

j-2  =  (c  +  xf  -{-y^  =  c^  +  2cx  +  x''  +  tf, 
and  7\^  =  (c  -  xf  ■i-if  =  c--2cx  +  x'^  -f-  y\  ^"^ 

By  subtraction,  we  obtain 

7-2  -  j\^  =  (r  —  ?'i)  (r  -f  rj)  =  4  ex. 
Hence,  since  r  +  r^  =  2  a, 

r^r,  =  y^  =  ^-^.  (3) 

2  a         a 
From  (1)  and  (3)  we  get. 


r  =  «  +  ^, 

(4) 

a 


Substituting  the  value  of  r  in  the  first  of  equations  (2),  we  obtain, 
after  reduction,  „  , 


x" 

o?-      a^  —  c^ 


A  further  simplification  is  obtained  by  putting 

a'~-c''  =  ¥,  (6) 

and  then  the  equation  assumes  the  final  form 

Equation  (7)  is  the  standard  form  of  the  equation  of  an  ellipse. 


^2       ..2 


a^     6^ 


62  LOCI  AND   THEIR   EQUATIONS  [Chap.  IV. 

50.  The  axes  and  eccentricity.  The  segment  of  the  line  joining 
the  foci  and  limited  by  tlie  curve  is  called  the  major  or  transverse 
axis  of  the  ellipse.  That  part  of  the  perpendicular  bisector  of 
the  segment  joining  the  foci  which  is  contained  within  the  curve 
is  the  minor  or  conjugate  axis  of  the  ellipse.  Thus,  AB  (Fig.  37) 
is  the  major  axis  and  CD  the  minor  axis.  The  axes  intersect  in 
the  center,  and  cut  the  curve  in  the  vertices. 

When  the  equation  of  the  ellipse  is  in  the  standard  form,  the 
axes  of  the  curve  coincide  with  the  axes  of  coordinates  (Art.  49). 
Hence  the  lengths  of  the  axes  of  the  ellipse  can  be  determined 
from  the  intercepts  (Art.  30)  made  by  the  curve  upon  the  coordi- 
nate axes.  From  equation  (7)  of  the  preceding  article  we  find 
that  the  intercepts  on  the  X-axis  are  ±  a  and  the  intercepts  on 
the  F-axis  are  ±  b.  Therefore  the  length  of  the  major  axis  is  2  a 
and  the  length  of  the  minor  axis  is  2  b.  The  segments  OB 
and  OD  (Fig.  37)  are  called  the  semimajor  axis  and  the  semi- 
minor  axis,  respectively. 

The  ratio  of  the  distance  between  the  foci  to  the  length  of  the 
major  axis  is  called  the  eccentricity  of  the  ellipse.  Since  the 
distance  between  the  foci  is  2  c  and  the  length  of  the  major  axis 
is  2  a,  the  eccentricity  is 

e  =  '-'  (1) 

a 


From  equation  (6)  (Art.  49),  c  =  Va^  —  b\     Therefore 

e  =  ^^^.  (2) 

a 

Since  a  is  always  greater  than  c,  the  eccentricity  of  the  ellipse  is 

necessarily  always  less  than  unity. 

Combining  equations  (4)  Art.  49,  with  equation  (1),  we  see  that 

the  lengths  of  the  focal  radii  of  the  point  P(x,  y)  are 

r  =  a  +  ex  and  ri  =  a  -  ex.  (3) 

EXERCISES 

1.  Find  the  equation  of  the  ellipse  for  which  the  sum  of  the  focal  radii  is 
8  and  the  distance  between  the  foci  is  6,  the  origin  being  at  the  center. 
What  is  the  eccentricity  of  this  ellipse  ?     Construct  the  ellipse. 

2.  An  ellipse  passes  through  the  points  (-  5,  0)  and  (0,  3)  and  is  sym- 
metrical with  respect  to  both  axes.  Find  the  coordinates  of  the  foci  and 
draw  the  curve. 


Arts.  50,  51] 


THE   HYPERBOLA 


63 


3.  "Write  the  standai'd  form  of  the  equation  of  the  ellipse  having 
given  :  (a)  the  length  of  the  transverse  axis  is  10  and  the  distance  be- 
tween the  foci  is  8  ;  (6)  the  sum  of  the  axes  is  18  and  the  difference  of  the 
axes  is  6 ;  (c)  transverse  axis  is  10  and  the  conjugate  axis  is  i  the  trans- 
verse axis  ;  (fZ)  transverse  axis  is  20  and  conjugate  axis  is  equal  to  the  dis- 
tance between  the  foci;  (e)  conjugate  axis  is  10  and  distance  between  the 
foci  is  10. 

v'i  9/2 

4.  The  equation  of  an  ellipse  is 1--^  =  1.     Find  the  lengths  of  the 

^  64      15 

focal  radii  of  the  points  whose  abscissa  is  \. 

5.  Find  the  lengths  of  the  semiaxes  and  the  eccentricity  of  each  of  the 
ellipses  whose  equations  are  : 


(a)  3  x2  +  2  2/2  =  6  ;  (6)    |  +  ^^ 


1 ;  (f )  X-  +  32/2  =  2;  {d)  42/^  +  2  z'^  =  2. 


6.  The  latus  rectum,  or  parameter,  of  an  ellipse  is  the  double  ordinate, 
or  double  abscissa,  passing  through  a  focus.  Find  the  length  of  the  latus 
rectum  for  each  of  the  ellipses  in  exercise  5. 

51,  The  h3rperbola.  A  point  moves  so  that  the  difference  of  its 
distances  from  ttvo  fixed  points  F  and  Fj^  is  constantly  equal  to  2  a. 
Construct  the  loctis  and  find  its  equation. 

Here  2  a  must  be  less  than  the  length  of  the  segment  FFi. 
For,  if  P  is  any  point  in  the  plane,  it  is  shown  in  geometry  that 
the  difference  be- 
tween any  two  sides 
of  the  triangle  PFF, 
is  less  than  the  third 
side. 

Lay  off  a  line  AB 
(Fig.  38)  2  a  units  in 
length  and  take  any 
point  C  on  this  line 
produced.  AVith  BC 
and  AC  as  radii  and 
F  and  F^^  as  centers, 
draw  arcs  of  circles 
intersecting,  in  M 
and  N.  These  points 
are  on  the  locus, 
since  the  difference 


Fig. 


B 

38 


64  LOCI   AND   THEIR  EQUATIONS  [Chap.  IV. 

of  the  radii  of  the  two  circles  is  2  a.  With  the  same  radii,  but 
interchanging  centers,  two  more  points,  M'  and  N',  are  obtained. 
Taking  G  at  different  places  on  AB  produced,  as  many  points 
on  the  locus  can  be  constructed  as  may  be  desired. 

The  locus  is  called  an  hyperbola,  the  points  F  and  jF\  are  its 
foci,  and  the  distances  from  any  point  on  the  curve  to  the  foci 
are  called  the  focal  radii  of  the  point.  The  two  parts  of  the  curve 
are  the  branches. 

To  find  the  equation  of  the  hyperbola,  we  proceed  as  in  the 
case  of  the  ellipse.  Let  the  line  joining  i^and  F^  be  the  X-axis, 
and  the  perpendicular  bisector  of  FF^,  the  I''-axis.  Let  F  be 
c  units  to  the  left  of  the  origin  and  F^,  c  units  to  the  right.  Take 
P{x,  y),  any  point  on  the  curve,  and  let  r  and  r^  be  the  lengths  of 
its  focal  radii  (r>?'i).     Then,  by  definition, 

/•  -  ri  =  2  a.  (1) 

Equations  (2)  of  Art.  49  hold  for  the  hyperbola,  and  we  obtain 
from  them,  by  subtraction, 

(r  —  ri)  (r  +  7\)  =  4  ex.  (2) 

Combining  (1)  and  (2),  we  have 

,  2  ex  /Q\ 

a 
From  (1)  and  (3)  we  get 

(4) 

a 

Substituting  the  value  of  r  in  the  first  of  equations  (2),  in  Art. 
49,  we  obtain,  after  reduction, 

2  2 

^ ^^— =  1.  (5) 

2  2 

a      c   —  a 

A  further  simplification  is  obtained  by  putting 

c2  -  tt"  =  W  (6) 

and  the  equation  assumes  the  final  form 

Equation  (7)  is  the  standard  form  of  the  equation  of  an  hyperbola. 


Art.  52]  AXES  AND   ECCENTRICITY  65 

52.  Axes  and  eccentricity.  The  hyperbola  meets  the  line  join- 
ing the  foci  iu  two  points  B^  and  A^  (Fig.  38)  which  are  equidis- 
tant from  the  mid-point  0,  as  may  be  seen  from  the  definition  of 
the  curve.  The  segment  B^A^  is  called  the  transverse  axis. 
Since  the  intercepts  on  the  X-axis  (when  the  equation  is  in  the 
standard  form)  are  ±  a,  the  length  of  the  transverse  axis  is  2  a. 

The  curve  does  not  meet  the  perpendicular  bisector  of  FF^, 
since  every  point  on  this  bisector  is  equidistant  from  i^'and  F^, 
but  a  segment  extending  b  units  above  0  and  6  units  below  O  is 
called  the  conjugate  axis.  0  is  the  center  of  the  curve,  and  the 
transverse  axis  meets  the  curve  in  the  vertices,  B^  and  A-^. 

The  ratio  of  the  distance  between  the  foci  to  the  length  of 
the  transverse  axis  is  called  the  eccentricity.  Since  FF^  =  2  c, 
and  B^Ai  =  2  a,  the  eccentricity  is 

^  =  --.  (1) 

From  (6),  Art.  51,  we  have  c  =  Va^-f  b%  and  therefore 

(2) 


^^^i^Tb' 


From  (1)  or  (2)  we  conclude  that  the  eccentricity  of  an  hyper- 
bola is  always  greater  than  unity. 

Combining   equations    (4)  of   Art.    51    Avith    equation    (1),  we 
have  the  lengths  of  the  focal  radii  in  terms  of  the  eccentricity, 

namely  : 

r  =  ex  +  a    and    r^  =  ex  —  a.  (3) 

EXERCISES 

1.  Write  the  standard  equation  of  the  hyperbola  for  which  the  difference 
between  the  focal  radii  is  6  and  the  distance  between  the  foci  is  8. 

2.  "Write  the  standard  equation  of  the  hyperbola  for  which  the  transverse 
axis  is  12  and  the  distance  between  the  foci  is  16. 

3.  Find  the  length  of  the  focal  radii  of  the  point  whose  ordinate  is  1  and 
whose  abscissa  is  positive,  the  equation  of  the  hyperbola  being ^  =  1 . 

4.  Find  the  semiaxes  and  eccentricity  of  each  of  the  hyperbolas  whose 
equations  are : 

(a)  4a;2-9?/2  =  36;   {h)   ^-^=1;  (c)   16  a-2  -  ?/2  =  16  ;  (d)    ^—-y'i=m. 


66 


LOCI  AND   THEIR  EQUATIONS  [Chap.  IV. 


5.  When  the  origin  of  coordinates  is  taken  at  the  center  of  an  hyper- 
bola and  the  foci  he  upon  the  T-axis,  the  standard  equation  is  - — ^  ==—  1. 

Find  the  le-ngths  of  the  semiaxes  and  tlie  eccentricity  of  each  of  the  follow- 
ing hyperbolas : 

(a)  3  2/2  _  2  X"  =  12  ;   (6)  4  x2  -  16  xfi  -  -64  ;  (c)  y"-  -  my?  =  n. 

6.  The  length  of  the  double  ordinate,  or  double  abscissa,  through  a  focus 
is  called  the  latus  rectum  of  the  hyperbola.  Find  the  length  of  the  latus 
rectum  for  each  of  the  hyperbolas  in  exercises  5  and  6. 


53.  The  parabola.  A  jyoint  moves  so  as  to  he  equally  distant 
from  a  fixed  point  and  from  a  fixed  straight  line.  Construct  the 
locus  and  find  its  equation. 

Let  F  be  the  fixed  point  and  AH  the  fixed  straight  line  (Fig. 

39).     Draw  AF  perpendicular  to  AH  and  a  series  of  lines  parallel 

to  AH,  as  PD,  P,D„  F^D^, 
etc.  With  AD  as  radius 
and  F  as  center,  draw  an 
arc  cutting  PD  in  P  and 
Q.  These  points  are  on 
the  locus,  since  PF=AD 
=  FQ.  E-epeating  the 
process  with  ADi,  AD^, 
etc.,  as  radii,  a  series  of 
points  on  the  locus  is  ob- 
tained. 

The  locus  is  called  a 
parabola  (cf.  Art.  28). 
The  fixed  point  F  is 
called  the  focus  of  the 
parabola    and    the    fixed 

line  AH  is  called  the  directrix.     The  point  0  is  the  vertex. 

To  find  the  equation  of  the  parabola,  let  AF  be  the  X-axis  and 

OY,  the  perpendicular  bisector  of  AF,  the  Y-axis.     Let  OF  —  p, 

and  P{x,  y)  be  any  point  on  the  curve.     Then  by  definition. 


H 

kY 

<F{x,y) 

,  P.. 

<^ 

y 

^ 

y 

y 

/ 

/ 

/f 

^; 

A 

1 

/ 

0 

F 

\ 

\ 

s 

S 

s 

S 

D 

~^X 

Fig.  39 


PF=AD. 


Abts.  53,  54]  THE   CASSINIAN   OVALS  67 


But  PF=^{x—py+if  and  AD  =  x+  p.     Therefore, 


VC^  —pf  +  f  =  X  +2h 
or  2/2  =  4.poc.  (1) 

Equation  (1)  is  tlie  standard  form  of  the  equation  of  the  parab- 
ola. The  number  p  is  called  the  parameter.  The  distance  from 
any  point  on  the  parabola  to  the  focus  is  called  the  focal  radius 
of  the  point.     The  length  of  the  focal  radius  of  any  point  (x,  y)  is 

r  =  x+p*  (2) 

EXERCISES 

1.  In  the  parabola  y-  =  ^x,  find  the  coordinates  of  the  focus  and  the  length  i  - 

of  the  focal  radius  from  the  point  (1,  2).  -4  j ■'  "  " 

2.  The  focus  of  a  parabola  is  at  the  point  (3,  0)  and  the  directrix  is  the    ' 

line  a;  + 1  =  0.     Find  the  equation.  ( 1 1  O) 

3.  The  focus  of  a  parabola  is  at  the  point  (0,  2)  and  the  directrix  is  the  ^ 
X-axis.     Find  the  equation.                                                                                         H     -^vt 

4.  If  the  focus  is  2  units  fi'om  the  vertex,  what  is  the  equation 
(a)  when  the  parabola  is  symmetrical  with  respect  to  the  X-axis  ? 
(6)   when  the  parabola  is  symmetrical  with  respect  to  the  J"-axis  ? 

5.  Construct  each  of  the  following  parabolas  : 

(a)  2/2  =8x;   (6)  y^  =  -Ax;  (c)  x:^  =  Qy;  (d)  x^=-lQy.  ^    -  y-(_-] 

V 

6.  The  double  ordinate,  or  double  abscissa,  through  the  focus  is  called  the 
latus  rectum  of  the  parabola.  Find  the  length  of  the  latus  rectum  of  each 
parabola  in  exercise  5. 

54.  The  cassinian  ovals.  A  point  moves  so  that  the  product  of 
its  distances  from  two  fixed  points  is  constantly  equal  to  a^.  Con- 
struct the  locus  and  find  its  equation. 

Let  F  and  F^  be  the  fixed  points  and  0  the  mid-point  between 

them  (Fig.  40).     Draw  the  circle  with  center  0  and  radius  OF, 

and  let  FM  be  the  tangent  to  this  circle  at  F.     Take  FM,  a  units 

in  length,  and  through  M  draw  a  series  of  secants  to  the  circle. 

Let  one  of  these  secants  meet  the  circle  in  the  points  A  and  A^. 

Then,  we  have  ^^,     ,,,        =-^         , 

'  MA  ■  MA^  =  FM  =  a2. 

Hence,  using  MA  and  MA^  as  radii  and  F  and  F-^  as  centers,  arcs 
of  circles  can  be  drawn  intersecting  in  points  of  the  locus.     Thus, 


68 


LOCI   AND   THEIR  EQUATIONS  [Chap.  IV. 


the  points  K,  L,  S,  and  T  are  on  tlie  locus.  Repeating  the 
process  with  other  secants,  as  many  points  of  the  locus  can  be 
constructed  as  may  be  desired. 

The  locus  is  called  a  cassinian  oval,  after  Cassini,  an  astronomer 
and  engineer  who  lived  in  the  latter  half  of  the  seventeenth 
century.     The  points  F  and  F^  are  the  foci. 


Fig.  40 


To  find  the  equation  of  the  locus,  let  FF^  be  the  X-axis  and  the 
perpeiidicular  bisector  of  FF^,  the  Y-axis.  Let  the  distance  be- 
tween the  foci  be  represented  by  2  c,  and  let  r  and  i\  represent 
the  focal  radii,  PF  and  PF^,  respectively.     Then,  as  in  Art.  49, 


r^  =  c-  +  2  ex  -f  x^  4-  y"^, 
7-j2  =  (.2  _  2  ex  -f  x^  +  y'^. 


(1) 


Multiplying  these  equations,  member  by  member,  and  remember- 
ing that  r  •  'i\  =  a^,  we  have 


or 


a^  =  (c2  +  x"  +  2/2)2  _  4  ^2.^-2, 

(.,.2  _|_  yy  _  2  C2  (.^2  -  2/2)  =  a'  -  C" 


(2) 


If  ((  =  c,  the  cassinian  oval  is  called  the  lemniscate.     This  is 
the  curve  shown  in  Fig.  40. 


Arts.  54,  56]    POLAR   EQUATION   OF  A  CIRCLE 


69 


EXERCISES 

1.  The  foci  of  a  cassinian  oval  are  at  the  points  (—2,  0)  and  (2,  0). 
Construct  the  curve  when  the  product  of  the  focal  radii  is  9  ;  when  the 
product  of  the  focal  radii  is  4 ;  when  the  product  of  the  focal  radii  is  1. 

2.  Find  the  intercepts  of  a  cassinian  oval  upon  the  coordinate  axes,  when 
a>  c,  when  a<^c,  and  when  a  =  c. 

3.  Show  that  a  cassinian  oval  is  necessarily  symmetrical  with  respect  to 
both  axes. 

65.  Recapitulation.  The  results  of  the  preceding  articles  are  so 
important  that  they  are  brought  together  here  in  compact  form. 
The  standard  forms  of  the  equations  should  be  memorized. 


Locus  OK  Curve 

Standard  Forms  of  the  Equation  in  Rectangular  Coordinates 

The  straight  line. 

(a)  Two-point  form ;      ^"^1  =  2/2-  2/i  _ 

^     ^                                                        X  —  Xi         X2  —  X\ 

(b)  Slope-point  form ;     y  —  yi  =  m  (x  —  Xi). 

(c)  Slope  form  ;                        y  =  rax  +  b. 

(d)  Intercept  form  ;         -  -|- 1  =  1. 

The  circle. 

(a)  (x-a)2-}-(2/-6)^  =  j-2. 
(&)  x2  +  y^  =  7-2. 

The  ellipse. 

X2         ^2_ 

a2  +  ft2      '■ 

The  hyperbola. 

x^      y2 

«2         6-2  -    '• 

The  parabola. 

?/2  =  4:pX. 

56.  Polar  equation  of  a  circle.  Let  C  =  (b,  a)  be  the  center  of  a 
circle  of  radius  a,  and  P=  {r,  0),  any  point  on  the  circle  (Fig.  41). 
In  the  triangle  COP,  we  have  OC  =  b,  OP=r,  and  the  angle 
COP  =  ±  {0  —  a),  depending  upon  the  position  of  P.  But  in 
either  case  the  law  of  cosines  applies  and  we  have 


r^  +  b^  -2  hr  cos(9  -  a)  =  a^ 


(1) 


70 


LOCI   AND   THEIR   EQUATIONS 


[Chap.  IV. 


nr,e) 


Fig.  41 


This  equation  expresses  the  relation  be- 
tween r  and  d  for  any  point  on  the 
circle  and  is,  therefore,  the -polar  equa- 
tion of  the  circle. 

If  the  initial  line  passes  through  the 
center  of  the  circle,  a  =  0  and  (1)  re- 
duces to 

r^  -I-  fe"  —  2  hr  cos  9  =  a^.  (2) 

If   the  pole   is   taken   on  the   circle, 


!"(>'. 0) 


Fig.  42 


b  =  a  and  (2)  becomes  the  im- 
portant form 

r  =  2a  cos  6.  (3) 

This  equation  is  also  immedi- 
ately deduced  from  Fig.  42, 
since  XPO  is  a  right  angle. 

If  the  pole  is  taken  at  the 
center,  b  —  0  and  (1)  becomes 
r  =  a,  (4) 

which  is  the  simplest  form  of  the  polar  equation  of  a  circle. 

57.    Polar  equation  of  a  straight  line.     Let  AB  be  any  straight 
line,  0  the  pole,  and  OX  the  initial  line  (Fig.  43).     Let  p  be  the 

length  of  the  perpendicular 
OM  let    fall    from   0   upon 
AB,  and  a  the  angle  XOM. 
r(r  0)  Take  P  (r,  6),  any  point  on 

the  line  AB.     Then 

^.         ,<  rcos(e-a)=_p        (1) 

is   the  polar   equation   of  the 
straight  line  AB. 


EXERCISES 

1.   The  center  of  a  circle  is  at 
the  point  whose  polar  coordinates 
are  i^,-\  and  the  radius  is  4.     AYrite  the  polar  equation  of  the  circle  and 


Fig.  43 


find  the  length  of  the  se£?ment  of  the  initial  line  within  the  circle. 


Arts.  57-59]    POLAR   EQUATION   OF  THE   ELLIPSE 


71 


2.  The  perpendicular  from  the  pole  upon  a  line  is  5  units  long  and  makes 
an  angle  of  60°  with  the  initial  line.  Write  the  polar  equation  of  the  line. 
With  origin  at  the  pole  and  X-axis  coinciding  with  the  initial  line,  write  the 
rectangular  equation  of  the  same  line  and  find  the  intercepts  on  the  axes. 

3.  A  circle  is  tangent  to  the  initial  line  at  the  pole,  its  radius  is  4  units 
long,  and  its  center  lies  above  the  initial  line.  What  is  the  polar  equation 
of  the  circle  ?  What  is  the  rectangular  equation,  the  origin  being  at  the 
pole,  and  the  X-axis  coinciding  with  the  initial  line  ? 

4.  Change  the  intercept  form  of  the  equation  of  a  straight  line  to  polar 
coordinates.     Show  that 


_    P 


P 


fr 


a  =  -^ —  and  b  = 

cos  a  sin  a, 

p  and  a  having  the  same  meanings  as  in  Art.  57. 

5.  Discuss  the  polar  equation  of  a  straight  line  (Art.  57)  for  a  =0^,  90°, 
180°.     Also  for  j3  =  0. 

6.  A  circle  passes  through  the  origin  and  has  its  center  on  the  line  bisect- 
ing the  first  and  third  quadrants.  Find  the  polar  equation  in  each  of  the 
two  possible  positions.     Also  the  rectangular  equation. 

58.  Polar  equation  of  the  parabola.  The  polar  equation  of  the 
parabola  assumes  the  simplest  form  when  the  pole  is  taken  at  the 
focus  and  the  initial  line  is  perpendicular 
to  the  directrix  (Fig.  44).  Let  P  {r,  9)  be 
any  point  on  the  parabola.  The  length 
of  the  focal  radius  PF  is  (Art.  53) 

r=  X  +  p.  (1) 

But,  from  the  figure, 

X  =  OD  =  r  cos  6  +p. 

Eliminating   x  between    (1)    and    (2)    and 
solving   the   resulting   equation    for   r,    we   have 

r  = 2p 

(1-  cosO) 


(2) 


Fig.  44 


(3) 


59.  Polar  equations  of  the  ellipse  and  the  hyperbola.  Take  the 
pole  at  the  left-hand  focus  and  the  initial  line  coincident  with 
the  transverse  axis  of  the  curve  (Figs.  45  and  46).  Then,  for 
either  curve,  the  length  of  the  focal  radius  PF  is  given  by  the 

formula  (Arts.  50  and  52) 

r  —  a  -\-  ex.  (1) 


72 


LOCI  AND   THEIR  EQUATIONS  [Chap.  IV. 


But,  from  either  figure, 
x=  OD  =  r  cos  6  -  c.  (2) 

Eliminating  x  between  (1) 
and  (2)  and  solving  the  re- 
sulting equation  for  r,  we 
have 

(g  -  ec) 


Fig.  45 


(1  —  e  COS  6) 
Replacing  e  by  its  value 


(3) 


(3)  becomes 


{a  —  ccosO) 


(4) 


For  the  ellipse,  a^  —  c^  =  If  (Art.  49)  and  a  >  c.  Hence  we  con- 
clude that  r  is  positive  for  all  values  of  6. 

For  the  hyperbola,  a^  —  c^  —  —  If  (Art.  51),  and  a  Kc  Hence 
a  —  c  cos  6   will    be    negative,   and    therefore   r   positive,    when 

cos  B'>  -    and  then   the   point 

c 

P{r,  6)  lies  on  the -right  branch 
of  the  curve.  For  example, 
when  ^  =  0,  cos  ^  =  1,  and 
T  —  a  +  c  =  FB.  The  expres- 
sion a  —  c  cos  6  will  be  posi- 
tive, and  therefore  r  negative, 

when  cos  ^  <  -  and    then    the 

c 

point  P(>-,  ff)  lies   on  the   left 

branch  of  the  curve.     For  example,  when  0 

and  r  =  —{c  —  a)  =  —  FA. 


180°,  cos  ^  =  -  1, 


EXERCISES 

1.  The  sum  of  the  focal  radii  is  8  and  the  distance  between  the  foci  is 
6.  Write  the  polar  equation  of  the  ellipse  and  sketch  the  curve  from  this 
equation. 

2.  The  difference  between  the  focal  radii  is  4  and  the  distance  between 
the  foci  is  6.  Write  the  polar  equation  of  the  hyperbola  and  sketch  the  curve 
from  this  equation. 


Arts.  59,  60]  PARAMETRIC   EQUATIONS  73 

3.  Show  from  the  polar  equation  that  the  radius  vector  for  the  ellipse 
is  always  finite  in  length. 

4.  Show  from  the  polar  equation  that  the  radius  vector  for  the  hyper- 
bola becomes  indelinitely  long  for  two  values  of  d,  each  less  than  180'^.  Find 
these  values. 

5.  The  focus  of  a  parabola  is  6  units  from  the  directrix.  Write  the  polar 
equation  and  sketch  the  curve  from  this  equation. 

6.  Show  from  the  polar  equation  of  the  parabola  that  the  radius  vector 
never  becomes  indefinitely  long  except  for  ^  =  2  rnt,  where  n  is  any  integer 
including  zero. 

7.  Show  that  the  polar  equation  of  the  lemniscate  is 

r-  =  2  c-  cos  2  d, 
the  pole  being  at  the  origin  and  the  initial  line  coinciding  with  the  A'-axis. 
Sketch  the  curve  from  this  equation. 

8.  Change  the  standard  forms  of  the  equations  of  the  ellipse,  hyperbola, 
and  parabola  to  polar  equations,  making  use  of  equations  (1),  Art.  9.  Why 
do  not  the  equations  thus  found  agree  with  the  polar  equations  in  Arts.  68 
and  59  ? 

9.  Derive  the  polar  equation  of  the  hyperbola,  assuming  the  right-hand 
focus  as  pole  and  the  transverse  axis  as  initial  line. 

10.  Derive  the  polar  equation  of  the  ellipse,  making  the  same  assump- 
tions as  in  the  preceding  exercise. 

11.  Compare    the    equation    r  = with   equation   4,    Art.   59. 

5  —  3  cos  d 

Does  the  given  equation  represent  an  ellipse  or  an  hyperbola  ?  What  is  the 
eccentricity  and  the  length  of  the  transverse  axis  ? 

12.  If  the  semiaxes  of  an  hyperbola  are  equal,  the  curve  is  called  the 
rectangular  hyperbola.  Write  the  polar  equation  of  the  rectangular 
hyperbola. 

60.  Parametric  equations.  It  is  frequently  useful  to  express 
the  X  and  y  coordinates  of  a  point  on  a  curve  in  terms  of  a  third 
variable  called  the  parameter.  For  example,  the  x  and  y  coordi- 
nates of  a  point  on  the  circle 

X^  -\-  y^z=  cO-  (1) 

can  be  expressed  as  follows : 

X  =  a  cos  t,     y=  a  sin  t,  (2) 

since  these  values  of  x  and  y  satisfy  (1),  whatever  value  is  given 
to  the  parameter  t.  Equations  (2)  are  parametric  equations  of  the 
circle  whose  equation  in  rectangular  coordinates  is  (1). 


74  LOCI  AND   THEIR   EQUATIONS  [Chap.  IV. 

Similarly,  a  pair  of  parametric  equations  of  tlie  ellipse 

x  =  a  cos  ^,     2/  =  0  sm  ^,  (4) 

since  these  values  of  x  and  ?/  satisfy  (3)  for  all  values  of  t. 

A  variety  of  pairs  of  parametric  equations  can  be  found  ex- 
pressing the  same  relation  between  x  and  y.     For  example, 

a(l-f)        T           2  at 
X  —  -^ ^  and  ?/  = 

are  parametric  equations  of  the  circle  (1),  since  these  values  of 
X  and  y  will  satisfy  (1)  for  all  values  of  t. 

EXERCISES 

1.  Show  that  X  =  a  sec  t  and  y  =  b  tant  are  parametric  equations  of  an 
hyperbola. 

2 .  Show  that  x  —  — \-t  and  y  =  b  +  mt  satisfy  the  slope  form  of  the 

m 

equation  of  a  straight  line  for  all  values  of  t. 

3.  Eliminate  t  from  the  equations  x  =  t-  and  y  =  2  t  and  thus  show  that 
these  equations  are  parametric  equations  of  a  parabola. 

4.  Write  a  pair  of  parametric  equations  for  the  standard  form  y'^  =  4px. 

5.  Show  that  x  —  at  and  y  =  b(l  —  t)    are  parametric  equations  of  a 
straight  line. 

6.  Show  that  the  equations 

cV2(l  +  fi)t 

1  +  «*       ' 

y  = !^ — 

1  +  t* 

are  parametric  equations  of  the  lemniscate  (Art.  54). 

7.  Write  the  parametric  equations  of  the  rectangular  hyperbola 

x^  —  y'^  =  a^. 

61.    Geometrical  construction  of   the  ellipse   and   the  hyperbola. 

The    ellipse    and  the    hyperbola    can   be   constructed   easily    by 
means  of  parametric  equations  (Figs.  47  and  48). 

Draw  the  concentric   circles  whose   radii  are  the  semiaxes  a 


Art.  61] 


GEOMETRICAL   CONSTRUCTION 


75 


and  h.     These   circles  are   called  the   major  and  minor  auxiliary 
circles,  respectively. 

For  the  ellipse,  with  any  value  of  t,  construct  OD  =  a  cos  t  and 
EP'  =  b  sin  t.  -  These  are  the  coordinates  of  the  point  P  on  the 
ellipse. 


/ 

fv^ 

■— ^ 

? 

/ 

>^ 

/\ 

/ 

\ 

X. 

\ 

y  / 

/ 

k 

^ 

\ 

\ 

\ 

\ 

\^^ 

A 

\ 

[X. 

\y 

X 

1 

\| 

3J 

V 

Fig.  47 


Fig.  48 


Similarly,  for  the  hyperbola,  OD  =  a  sec  ^  and  EM=h  tan  ^ 
are  the  coordinates  of  the  point  P  on  the  curve. 

The  points  P,  P,  and  P"  are  called  corresponding  points.  As 
the  radius  OP  revolves  about  0,  the  points  P  and  P"  move  on 
their  respective  auxiliary  circles,  and  P  describes  the  ellipse  in 
Fig.  47  and  the  hyperbola  in  Fig.  48. 

EXERCISES 

1.  Write  the  parametric  equations  and  coustruct  the  ellipse  whose  semi- 
axes  are  3  and  4. 

2.  "Write  the  parametric  equations  and  construct  the  hyperbola  whose 
semiaxes  are  3  and  4. 

3.  Construct  the  following  loci  by  assigning  arbitrary  values  to  the 
parameter  t  and  tabulating  the  corresponding  values  of  x  and  ?/ : 

(a)  x  =  «-l,  ?/  =  4-<2;    (J)-)  x  =  ~,  y  =  -;    (c)  z  =  St,  y -3f^  -  t^. 

4.  With  e  as  the  parameter,  construct  the  locus 

a;  =:  6  cos  61  +  3  cos  2  6, 
2/  =  6  sin  6*  —  3  sin  2  0. 
This  locus  is  called  the  three-cusped  hypocycloid. 


76  LOCI   AND   THEIR   EQUATIONS  [Chap.  IV. 

62.    Recapitulation. 


Locus  OR  Curve 

Polar  Equation 

Parametric  Equations 

The  circle. 

r  =  2  a  cos  ^. 

X  =  a  COS  t,  y  =  a  sin  t. 

The  ellipse. 

«^  -  c-^     _  ^ 

a>  c. 

X  =  a  COS  t,  y  =  b  sin  t. 

a  —  c  cos  ^ 

The  hyperbola. 

ff  —  c  cos  6 

a<_c. 

x  =  a  sec  t,  y  =  b  tan  t. 

The  parabola. 

2;j      _,. 

4p 

1  —  cos  ^ 

EXERCISES 

1.    Find  the  lengths  of  the  axes,  the  distance  between  the  foci,  and  the 
eccentricity  of  each  of  the  following  cnrves. 


(a)  9  2/2  +  4  a;2  =  36. 

(c)   100  2/2 -25x2  ==-2500. 

(e)  64  2/2  +  25  x^  =  1600. 

2.    Show  that    the    points    (—  4, 
(2,  —  1)  lie  upon  two  straight  lines. 


(6)  7  x2  +  11 2/2  =  15. 
((?)   17x2-25  2/2  =  -  116. 
(/)  64  2/2 -25x2  =-1600. 

-2),    (2,   1),    (-6,   3),    (0,0),    and 
What  are  the  equations  of  these  lines  ? 


3.  The  semiaxes  of  an  ellipse  are  6  and  4.  Find  the  length  of  the  latus 
rectum. 

4.  Write  the  polar  equation  of  the  hyperbola,  if  the  transverse  axis  is  6 
and  the  distance  between  the  foci  is  10.     For  what  values  of  6  is  r  infinite  ? 

5.  If  the  perpendicular  to  the  major  axis  of  an  ellipse  at  the  point  D 
meets  the  major  auxiliary  circle  in  P  and  the  ellipse  in  P',  prove  that 

DP :  DP'  -.-.a-.b, 

where  a  and  b  are  the  semiaxes. 

6.  In  geometry  it  is  shown  that  the  areas  of  rectangles  having  the  same 
width  are  to  each  other  as  their  lengths.  Combining  this  proposition  with 
that  in  the  preceding  exercise,  show  that  the  area  of  the  major  auxiliary 
circle  is  to  the  area  of  the  ellipse  as  a  is  to  b,  and  hence  the  area  ofthe 
ellipse  is  Trab. 

7.  If  the  major  auxiliary  circle  is  rotated  around  the  major  axis  of  the 

ellipse  until  its  plane  makes  an  angle  whose  cosine  is  —  with  the  plane  of 

a 

the  ellipse,  and  if  perpendiculars  be  dropped  from  every  point  of  the  circle 

upon  the  plane  of  the  ellipse,  show  that  the  feet  of  these  perpendiculars  lie 

upon  the  ellipse. 


CHAPTER  V 
EQUATIONS  AND   THEIR  LOCI 

63.  Locus  of  an  equation.  The  curve  which  passes  through 
all  the  points  whose  coordinates  satisfy  a  given  equation,  and 
through  no  other  points,  is  called  the  locus  of  the  given  equation. 

64.  A  second  fundamental  problem.  In  the  last  chapter  we 
have  found  the  equations  of  a  number  of  important  loci  from 
given  laws.  There  now  arises  a  second  fundamental  problem  of 
analytic  geometry ;  namely,  given  an  equation  connecting  the  varia- 
bles X  and  y,  to  construct  the  locus  of  the  equation  and  to  discover  the 
important  properties  of  the  locus. 

In  simple  cases  the  general  form  of  the  locus  can  be  determined 
by  plotting  (Art.  27).  But  this  method  alone  often  fails  to  reveal 
the  important  properties  of  the  locus,  and,  at  best,  leaves  wholly 
undetermined  the  form  of  the  locus  between  consecutive  points 
plotted.  A  discussion  of  the  given  equation,  however,  will 
reveal  certain  properties  of  the  locus  which,  together  with  a  few 
plotted  points,  will  determine  frequently  the  form  and  nature  of 
the  locus. 

65.  Discussion  of  an  equation.  The  method  to  be  followed 
must  depend  upon  the  particular  equation  under  discussion,  but 
the  following  outline  will  serve  to  indicate  what  to  look  for  in 
any  given  case. 

(a)  Symmetry  (cf.  Art.  29). 

(1)  If  the  given  equation  contains  only  even  powers  of  y,  the 
locus  is  symmetrical  with  respect  to  the  X-axis.  For  then,  if 
P(a,  b)  is  any  point  on  the  locus,  Q(a,  —  b)  is  also  on  the  locus. 

(2)  If  the  given  equation  contains  only  even  powers  of  x,  the 
locus  is  symmetrical  with  respect  to  the  I^axis.  For  then,  if 
P(a,  b)  is  any  point  on  the  locus,  Q(—a,  b)  is  also  on  the  locus. 

(3)  If  the  given  equation  contains  only  even  powers  of  x  and  of 


78  EQUATIONS  AND  THEIR  LOCI  [Chap.  V. 

y,  the  locus  is  symmetrical  with  respect  to  the  origin.  For  then, 
if  P(a,  6)  is  any  point  on  the  locus,  Q(— a,  —6)  is  also  on  the 
locus. 

(4)  If  the  given  equation  is  unaltered  when  x  and  y  are  inter- 
changed, the  locus  is  symmetrical  with  respect  to  the  line  bisect- 
ing the  first  and  third  quadrants.  For  then,  if  P{a,  h)  is  any 
point  on  the  locus,  Q(h,  a)  is  also  on  the  locus  (cf.  Art.  39). 

(6)  Intercepts  (cf.  Art.  30).  The  determination  of  the  inter- 
cepts furnishes  a  good  point  from  which  to  begin  the  construction 
of  the  locus. 

(c)  Limits  of  the  locus.  It  frequently  happens  that  to  certain 
values  of  either  variable  there  correspond  no  real  values  of  the 
other.  There  is  no  corresponding  real  point  in  such  a  case. 
Hence,  tJie  locus  is  confined  to  those  prxrts  of  the  plane  such  that  to 
each  value  of  either  variable  there  corresponds  a  real  vahie  of  the 
other.  For  example,  the  locus  of  the  equation  y^  =  2  x  (Art.  28) 
is  confined  to  the  part  of  the  plane  to  the  right  of  the  T'-axis. 

Whenever  to  any  value  of  either  variable  there  corresponds  no 
real  value  of  the  other,  the  locus  is  said  to  be  limited.  The  equa- 
tion of  a  limited  locus,  if  it  is  algebraic,  must  establish  at  least 
one  of  the  variables  as  a  multiple-valued  function  of  the  other. 
For  in  no  other  way  can  imaginary  values  appear.  The  converse 
does  not  hold,  for  an  equation  which  establishes  one  variable  as  a 
multiple-valued  function  of  the  other  does  not  necessarily  have  a 
limited  locus.  Thus,  the  equation  y-  —  x"^  establishes  ?/  as  a 
multiple-valued  function  of  x,  but  for  no  value  of  either  variable 
is  the  other  ever  imaginary. 

(rf)  Change  of  one  variable  due  to  a  given  variation  of  the  other 
(Art.  27).  It  is  important  to  determine  from  the  equation  how 
increasing  or  decreasing  one  variable  will  affect  the  other.  For 
example,  if  x  is  allowed  to  increase  in  value,  will  y  increase  or 
decrease  in  value  ?  In  other  words,  to  determine  whether  ?/  is  a 
monotone  function  or  not ;  and,  if  not,  for  what  values  of  x  it  is 
increasing,  for  what  values  it  is  decreasing,  and,  if  possible,  for 
what  values  of  x  it  has  turning  points. 

(e)  Behavior  of  the  locus  at  great  distances  from  the  origin. 

It  is  also  important  to  determine  from  the  equation  how  in- 
creasing either  variable  indefinitely  will  affect  the  other. 


Art.  66] 


EXAMPLE   I 


79 


The  discussion  of  an  equation  according  to  the  foregoing  out- 
line will  be  illustrated  in  the  following  examples. 

66.  Example  I.  Discuss  the  equation  x'^  +  4  ?/"'  =  4  and  construct  tlie 
locus. 

(a)  Assume  any  point  P{a,  b)  whose  coordinates  satisfy  the  given  equa- 
tion (Fig.  49).     Then  the  coordinates  of  the  points  Q(a,  —  b),  S{~  a,  6), 


Fig.  49 


and  i?(—  ff,  —  b)  also  satisfy  the  equation.     Hence  the  locus  is  symmetrical 
with  respect  to  both  axes,  and  also  with  respect  to  the  origin. 

(b)  The  x-intercepts  are  ±  2  and  the  2/-intercepts  are  ±  1. 

(c)  Solving  the  equation  for  y,  we  have 


y 


V4  —  x^ 


(1) 


from  which  it  is  seen  that  x  is  limited  to  the  range  of  values  extending  from 

—  2  to  +  2  in  order  that  y  may  have  real  values.     The  range  of  values  for  x 

is  indicated  by  writing 

-2g.>-^  +  2. 


The  locus  is  thus  limited  to  lie  between  the  lines  x  = 
lines  AB  and  CD  in  the  figure. 

Again,  solving  the  equation  for  x,  we  obtain 


a;=±2Vl 


r 


—  2  and  x  =  +  2,  or  the 


(2) 


Hence  y  is  limited  to  the  range 


-l^y<l 

in  order  that  x  may  have  real  values.  The  locus,  therefore,  lies  between  the 
lines  y  =  +1  and  ?/  =  —  1,  or  the  lines  BC  and  AD  in  the  figure.  The  locus, 
therefore,  lies  wholly  within  the  rectangle  ABC'D. 

(d)  From  equation  (1)  it  follows  that  as  x  increases  or  decreases  from 
zero,  the  positive  value  of  y  decreases,  and  the  negative  value  increases. 


80 


EQUATIONS   AND   THEIR  LOCI  [Chap.  V. 


Hence  we  conclude  that  +  2  is  a  maximum  value  of  ?/,  and  —  2  is  a  minimum 
value. 

Similar  conclusions  with  respect  to  the  values  of  x  can  be  drawn  from 
equation  (2). 

The  foregoing  discussion  reveals  the  general  form  and  properties  of  the 
locus.     The  curve  is  an  ellipse. 


67.  Example  II.  Discuss  the  equation  x^  — 4  2/2  =  4  and  find  the  form 
and  general  properties  of  the  locus. 

(a)  The  locus  is  symmetrical  with  respect  to  each  axis  and  with  respect  to 
the  origin  as  in  Example  I. 

(&)  The  intercepts  on  the  X-axis  are  ±  2  ;  the  locus  does  not  meet  the 
F-axis. 

(c)  Solving  the  equation  for  ?/,  we  have 


y 


±  Vx^  —  4 


(1) 


Here  y  is  imaginary  for  all  values  of  x  within  the  range  from  —  2  to   +2. 

Hence  the  ranse  for  x  is 

-  2  >  x  >  2 


:^  "J../  .    .  ;^:    . : 

;  :~              -    '\  l^' 

■-■■--.    -^' 

i —   :  ,  :       :  M  i 

_j_._^...  - 

-^S5^:M^ 

-— 

-;'- 

_X UL 

1 M-^^ 

_ ^_ ! 

"  ;    """"^      ^    ~X 

---'i'f'iiw 

LJ-1^ 

=-=;— ^ — -  — 

K_p!r 

Wn 

^ 

—/ 

III 

I^^T^^^ 

-■=^ 

i^v^--- 

"^^      \ 

====^^i 

--H-+t  — -^1 



>~v^^^ 

' '"'""r"""Tli  " .": "' ^ ' 

-     :-■;    ■ 

1  j-  ^" 

— -+ 

-|--|--+-|--^-r|-|-|--|-|-| H 

l- 

-H 

y: 

I-4---I4-I-- 

Fig.  50 


In  order  that  rj  may  have  real  values.     The  locus,  therefore,  lies  outside  the 
strip  bounded  by  the  lines  x  =  —  2  and  x  =  +  2  (Fig.  50). 
Solving  the  equation  for  x,  we  have 

X  =  ±  2  Vl  +  2/-. 


Arts.  67,  68]  EXAMPLE   III  81 

Hence  x  is  real  for  all  values  of  y.  The  locus  is  therefore  unlimited  in  the 
y-direction. 

{d)  From  equation  (1),  as  x  increases  from  2,  the  positive  value  of  ij 
increases  continually  and  %Yithout  limit,  but  as  x  increases  from  a  large 
negative  value  to  —  2,  the  positive  value  of  y  continually  decreases  to  zero. 
Combining  these  facts  with  the  symmetry  in  («)i  "we  conclude  that  the  locus 
spreads  out  as  it  recedes  from  the  origin  in  either  direction. 

(e)  As  X  increases  indefinitely,  the  values  of  y  approach  nearer  and  nearer 


to  ±  -.     For  the  radical  Vx-  —  i  is  clearly  always  less  than  x  in  value,  but 
for  very  great  values  of  x,  the  difference 


_  ^yO.  _  4  =:  . 


X  +  Vx-  —  4 

is  very  small  and  can  be  made  as  small  as  we  please  by  choosing  x  sufficiently 
great.  Therefore,  at  great  distances  from  the  origin,  the  locus  lies  close  to 
the  straight  lines 

y=±ry 

These  lines  are  called  asymptotes.  The  X-axis  bisects  one  of  the  angles 
between  the  asymj^totes  and  the  locus  lies  within  this  angle,  one  branch  on 
each  side  of  the  origin.     The  curve  is  an  hyperbola. 

EXERCISES 

1.  Discuss  the  following  ecpiations  and  draw  the  corresponding  loci : 

(a)  4  x2  +  9  y2  =  36.  (6)  4  a;2  -  9  2/2  =  36.  (c)  y^  =  16  x.  {d)  x^  =  dy. 
(e)  a;2  -1/2^4.  (/)  a;2  +  ?/2  =  4.  (g)  ?/  =  4  x^. 

2.  Find  the  lengths  of  the  axes,  the  distance  between  the  foci  and  the 
eccentricity  of  the  ellipse  in  exercise  1. 

3.  Find  the  lengths  of  the  axes,  the  distance  between  the  foci,  and  the 
eccentricity  of  the  two  hyperbolas  in  exercise  1. 

4.  Find  the  equations  of  the  asymptotes  for  each  of  the  hyperbolas  in  ex- 
ercise 1. 

5.  Find  the  coordinates  of  the  focus  for  each  of  the  parabolas  in  exer- 
cise 1. 

68.  Example  III.  Discuss  the  equation  xy  —  x  —  y  =  0  and  find  the 
form  and  propeities  of  the  locus. 

(a)  The  locus  is  obviously  not  symmetrical  with  respect  to  either  axis  nor 
with  respect  to  the  origin.  It  is,  however,  symmetrical  with  respect  to  a  line 
bisecting  the  first  and  third  quadrants,  since  if  P(«,  6)  is  any  point  whose 
coordinates  satisfy  the  equation,  then  the  coordinates  of  the  point  ^(6,  «) 


82 


EQUATIONS  AND   THEIR   LOCI 


[Chap.  V. 


also  satisfy  the  equation.    The  points  P  and  Q  are  symmetrically  situated 
with  respect  to  the  line  OA  (Fig.  51). 

(5)  The  locus  crosses  the  coordinate  axes  only  at  the  origin.     The  inter- 
cept on  each  axis  is  therefore  zero. 


Fig.  51 

(c)  The  locus  is  not  limited  in  either  direction,  since  each  variable  is  real 
for  all  values  of  the  other. 

(d)  Solving  the  equation  for  y,  we  have 


2/  = 


(1) 


Hence,  as  x  increases  from  a  large  negative  value,  y  continually  decreases 
from  a  value  less  than  1,  through  zero,  becoming  —  co  for  x  equal  to  1.  As 
X  passes  the  value  1,  y  changes  suddenly  to  a  very  great  positive  value  and 
then  continually  decreases,  approaching  nearer  and  nearer  to  1.  The  func- 
tion y  is  therefore  monotone.     It  has  a  discontinuity  at  a;  =  1. 

(e)  From  equation  (1)  we  see  that  y  approaches  nearer  and  nearer  to  1 
as  X  increases  or  decreases  indefinitely.  For  very  great  values  of  x,  there- 
fore, the  locus  lies  close  to  the  line  y  =  1.  This  line  is  an  asymptote  to  the 
curve. 

Similarly,  solving  the  equation  for  ,r,  the  line  a:  =  1  is  found  to  be  an 
asymptote  ;  that  is,  for  very  great  values  of  y  the  locus  lies  close  to  this  line. 

The  above  discussion  enables  us  to  form  a  fairly  accurate  idea  of  the  locus 
before  any  plotting  has  been  done. 


69.   Example  IV.      Discuss  the  equation  y 
and  properties  of  the  locus. 


l  +  a;^ 


and  find  the  form 


Arts.  69,  70] 


EXAMPLE  V 


83 


(a)  The  locus  is  not  symmetrical  with  respect  to  either  axis,  but  it  is 
symmetrical  with  respect  to  the  origin,  since  if  P(a,  b)  is  any  point  on  the 
locus,  so  also  is  the  point  Q(—  a,  —  b)  on  the  locus  (Fig.  52). 


..i\7- 

j_ 

[                                                          1 

1      ■           1      1            1 

i                                                              •                         i  '     I  1  1   1 

:                 1         ;    ■                             >                                                                                      ,    ■     1        '        ■ 

L_i_^ LL^3_ "]  1  -  -L       ^_ 

/■  1  1 

/         ^                                                         \ 

-." --f-J L_U_Lj.-aJJ ^  Q^ ^'--1 L_LJ__L^__Jj -J--  —  -^ 

1    '         1     1    1    1        ;         '         X^     '                :     ,    1        1     1                 ■         [            I 

.1                                                           1   M       1    1   i 

Fig.  52 


(6)  The  locus  crosses  the  axes  only  at  the  origin. 

(c)  The  locus  is  unlimited  in  the  a:-direction,  since  y  is  real  for  all  values 
of  X.     But  if  we  solve  the  equation  for  x,  we  obtain 


2y 

and  therefore  y  is  limited  to  the  range 

in  order  that  x  may  have  real  values.    Consequently  the  locus  lies  within 
the  strip  bounded  by  the  lines  ?/  =  —  |  and  ?/  =  +  i. 

(d)  and  (e).  As  x  increases  from  zero  to  1,  y  increases  from  zero  to  ^  ; 
and  as  x  increases  indefinitely  from  1,  y  decreases  slowly  from  i  towards 
zero.  Hence  the  function  y  has  a  turning  point  at  x  =  1  and  its  value  there 
is  |.  Also  for  very  great  positive  values  of  x,  the  locus  lies  close  to  the 
X-axis.  Since  the  locus  is  symmetrical  with  respect  to  the  origin,  its  form 
to  the  left  of  the  origin  is  known  as  soon  as  its  form  to  the  right  has  been 
determined.  We  conclude,  therefore,  that  the  function  has  a  turning  point 
at  X  =  —  1  and  that  the  X-axis  is  an  asymptote  to  the  curve. 

70,   Example  V.     Discuss  the  equation  y^  —  ^ —     ^'^  and  find  the  form 


and  properties  of  the  locus. 


3  a;  -f  6 


84 


EQUATIONS  AND  THEIR  LOCI 


[Chap.  V. 


(a)  The  locus  is  clearly  symmetrical  witli  respect  to  the  X-axis  ;  it  is  not 
symmetrical  with  respect  to  the  F-axis,  since  the  equation  contains  odd  powers 
of  X. 

(b)  For  the  purpose  of  discussion  we  will  suppose  that  &  is  a  positive 
number  (Fig.  53).  The  locus  crosses  both  axes  at  the  origin  and  also  crosses 
the  X-axis  at  x  =  ft. 


1  1  1  1  1  1  1  1  1 

iijj  ;:i: 

y  ' 

'  ■  ■  i_ —  • 

.   :     -   y 

+hr  r : 

t 

H 

^::\    ;: 

-ii 

Kv. 

^ 

-_ 

fi^^ 

:::? 

=::■  ;:;■ 

t. 

J:  :::::::::::;:::::::::: 

"  1 1  "  1 

-a^m- 

,::. 

mf 

llfl::::;::::::::::::::::::::: 

Fig.  53 


(c)  From  the  equation  we  see  that  x  is  limited  to  the  range 

-^<x<b 
3=    = 

in  order  that  y  may  have  real  values.     The  locus  therefore  lies  between  the 

lines  a;  =  —  -  and  x  =  b. 
8 

((?)  As  X  increases  from  zero  to  b,  the  absolute  value  of  y  at  first  increases 

and  then  decreases  to  zero.     This  shows  that  the  locus  has  a  loop  at  the  right 

of  the  origin.     As  x  decreases  from  zero  to  — ,  the  absolute  value  of  y  in- 

o  " 

creases  very  rapidly  from  zero,  becoming  infinite  at  x  =  —  -  •    The  line  a:=  —  - 

b  O 

is  an  asymptote  to  the  curve.    The  locus  is  called  the  folium  of  Descartes. 


EXERCISES 

1.    Discuss  the  following  equations  and  plot  the  corresponding  loci.     Find 
the  asymptotes  when  these  exist. 


Art.  71] 


EXAMPLE   VI 


85 


(a)  xy  —  x  +  y-Q. 
(d)x22/2  =  (2/  +  2)-^(9-2/2). 


(&)  2/2  _  4^.2 
{e)y=-    "" 


4. 


(c)  x^  +  2/2  =  a|. 
(/)2/=        ^ 


1— x-  ""  "       (x  —  2)2 

2.  Find  the  lengths  of  the  semiaxes,  the  coordinates  of  the  foci,  and  the 
eccentricity  of  the  hyperbola  whose  equation  is  (5)  of  the  previous  exercise. 

Note.     The  locus  (d)  in  exercise  1  is  called  the  conchoid  of  Nicomedes. 
71.   Example    VI.     Discuss    the    equation    of  the  catenary;    namely, 
?/  =  -  (e"  +  e    "),  and  plot  the  locus. 

The  equations  discussed  in  the  foregoing  examples  are  algebraic  equations 
and  the  corresponding  loci  are  called  algebraic  curves.  The  locus  of  a  tran- 
scendental equation  is  called  a  transcendental  curve.  Thus  the  catenary  is  a 
transcendental  curve. 

(a)  The  locus  is  symmetrical  with  respect  to  the  T-axis,  since  changing 
the  sign  of  x  does  not  alter  the  equation.  The  locus  is  not  symmetrical  with 
respect  to  the  X-axis,  since  for  every  value  of  x,  y  is  positive.  The  curve 
therefore  lies  entirely  above  the  X-axis. 

(6)  The  curve  meets  the  F-axis  a  units  above  the  origin.  It  does  not  meet 
the  X-axis. 

(c)  The  locus  is  un- 
limited in  the  x-direction, 
since  y  is  real  for  every 
value  of  X. 

{d)  and  (e)  As  x  in- 
creases  from   zero,    y   also 

X 

increases,   since  neither  Ca 

nor  e~ a  can  ever  become 
negative.  We  conclude, 
therefore,  that  the  function 
y  has  a  turning  point  at  the 
origin  and  that  its  value 
there  is  a  minimum. 

To  plot  the  locus,  we  con- 
struct the  auxiliary  curves 

2/1  =  en  and  2/2  =  e~« 
as  in  Art.  37,  taking  a  for 

the  unit  of  measure.     These  curves  are  respectively  AB  and  A'B'  (Fig-  54). 
The  required  locus, 

y 


vi  +  m 


bisects  the  segment  of  each  ordinate  contained  between  the  auxiliary  curves. 


86  EQUATIONS  AND   THEIR   LOCI  [Chap.  V. 

The  catenary  is  the  curve  formed  by  a  flexible  chain  hung  between  two 
supports.  It  is  of  great  importance  in  problems  connected  with  the  con- 
struction of  suspension  bridges. 

72.  Simple  harmonic  curves,  compound  harmonic  curves.  The 
loci  of  the  equations 

■       2   TVX  T  2   TTX 

y  =  a  sin and  ?/  =  a  cos 

'^  rp  ^  rp , 

are  called  simple  harmonic  curves.  They  are  constructed  as  in 
Art.  35.  Simple  harmonic  curves  represent  simple  vibratory 
or  wave  motion  like  that  of  a  swinging  pendulum  carried  forward 
with  a  uniform  velocity  in  a  straight  line  perpendicular  to  the 
plane  in  which  the  pendulum  is  swinging.  The  number  a  is 
called  the  amplitude  of  the  vibration,  and  T,  the  period  (cf. 
exercises,  Art.  35). 

The  loci  of  equations  of  the  form 

.     2  ttX    ,    -,      .     2  ttX  ■     2  ttX    ,    ,  2  ttX 

w  =  a  sm ±  0  sm or  w  =  a  sm ±  b  cos 

are  called  compound  harmonic  curves.  To  construct  a  compound 
harmonic  curve,  we  plot  each  of  the  simple  harmonic  curves  of 
which  it  is  composed  on  the  same  coordinate  axes  and  take  the 
algebraic  sum  of  the  ordinates  for  any  particular  value  of  x  as  the 
ordinate  of  the  required  curve  for  that  value  of  x. 

In  general,  to  construct  the  locus  of  an  equation  of  the  form 

2/=/l(^)±/2(«), 

plot  each  of  the  auxiliary  curves 

2/i=/i(^)  and  y2=Mx) 

on  the  same  coordinate  axes  and  take  the  algebraic  sum  of  the 
ordinates  for  any  particular  value  of  x  as  the  ordinate  of  the 
required  locus  for  that  value  of  x. 

Compound  harmonic  curves  occur  in  the  theories  of  sound, 
light,  and  electricity.  Several  simple  harmonic  curves  may  be 
combined  to  form  a  compound  harmonic  curve. 


Arts.  72,  73] 


DAMPED  VIBRATIONS 


87 


EXERCISES 

1.  If  a  pendulum  makes  4  complete  vibrations  per  second,  show  that  its 
period  is  T  —  I.  If  the  amplitude  of  the  vibration  is  2,  show  that  the  motion 
of  a  point  on  the  pendulum  is  given  by  the  equation  «/  =  2  sin  8  ttx,  where  x 
represents  time  measured  in  seconds.     Construct  the  locus  of  the  equation. 

2.  Construct  the  loci  of  the  following  equations  : 

(a)  y  —  e''  +  sin  x.  (6)  ?/  =  a;  +  sin  x.  (c)  y  =  2  .r  — cos  x. 

{d)  y  =  sinx  +  sin  2  x.  (e)  y  =  x^  +  2''.  (/)  y  =  x  —  sm2  x. 

3.  The  piston  of  an  engine  is  connected  to  the  drive  wheel  by  a  connecting 
rod.  If  the  crank  pin  describes  a  circle  whose  radius  is  2  feet  and  makes  200 
revolutions  per  second,  what  is  the  amplitude  and  the  period  of  the 
harmonic  motion  described  by  the  piston  ?  Write  the  equation  expressing 
this  motion. 

73.    Damped  vibrations.     The  loci  of  equations  of  the  form 

y  =  ae''''^  sin  kx  ov  y  =  ae'""'"  cos  hx 

represent  damped  vibrations  such  as  a  pendulum  vibrating  in  a 


1 

/ 

} 

A 

^ 

^ 

-, 

^ 

— ■ 

"^ 

— 

- 

. 

/" 

\ 

' 

— 

— ■ 

— 





= 

/ 

\ 

" 

— 

-J 

^ 

Ji 

/ 

\ 

^ 

^ 

^ 

0 

\ 

\ 

3 

/4 

5 

^  1 

S 

V, 

^ 

X' 

_ 

— 

— 

■ — 

'n' 

j:=: 

— ' 

__ 



— 

" 

L^ 

-A 

7 

L_ 

Fig.  55 


resisting  medium.     To  illustrate  the  method  of  plotting  the  loci 
of  such  equations,  we  will  construct  the  locus  of  the  equation 


-J^      •       TVX 

e  *  sm  — 

2 


88  EQUATIONS   AND  THEIR   LOCI  [Chap.  V. 

Since  the  absolute  value  of  tlie   sine  can  never  exceed  unity, 
we  see  that  the  absolute  value  of  y  can  never  exceed  the  value  of 

e~^.     Again,  when  x  is  any  odd  integer,  sin  —  is  either  + 1  or 

—  1,  and  when  x  is  an  even  integer,  sin  —  is  zero.     Hence  we 
conclude  that  the  required  locus  lies  between  the  two  curves 

y  —  e"'""  and  y  =  —  e"^""  (1) 

and  crosses  the  X-axis  whenever  x  is  an  even  integer. 

The  two  curves  in  (1)  can  be  constructed  as  in  Art.  37,  and  are 
called  boundary  curves.  The  locus  is  shown  in  Fig.  55,  where  AB 
and  A'B'  are  the  boundary  curves. 

In  general,  the  loci  of  equations  of  the  form 

y  =f(x)  sin  kx  or  y  =f(x)  cos  kx 

can  be  constructed  by  first  plotting  the  loci  of  the  boundary  curves 

y=f(x)  and  ?/ = -/(a;). 

EXERCISES 

1.    Construct  the  following  loci : 

(a)  y  =  xs\nx.  (5)  y  =  xcosx. 

■(c)  y  —'-  sin (d)  y  =  -  sin  x. 

o  o  X 

(e)  y  —  x^sinx.  (/)  y  =  e'^'sinx. 

TTX 


(g')  y  =  e^  sin  x.  (h)  y  =  (3  +  ^  j  sii 

2.   Discuss  the  equation  y'^  —  xsivfi  x  and  construct  the  locus. 


sin 

2 


74.  Polar  equations.  When  the  given  equation  is  in  polar 
coordinates,  the  main  facts  about  the  locus  can  also  be  determined 
by  a  discussion  of  the  equation.  The  points  to  be  determined  by 
the  discussion  are  the  following : 

(a)  Symmetry  tvith  respect  to  the  pole. 

(1)  The  locus  is  symmetrical  with  respect  to  the  pole  if,  for 
any  given  value  of  B,  the  equation  is  satisfied  by  both  +  r  and 
—  r.  This  will  happen  when  the  equation  contains  only  even 
powers  of  r. 


Arts.  74,  75] 


EXAMPLE   IX 


(2)  The  locus  is  symmetrical  with  respect  to  the  pole  if,  when- 
ever the  equation  is  satisfied,  by  a  point  (t;  9),  it  is  also  satisfied 
by  (r,  ^  +  180°).  For  then  the  locus  cuts  each  radius  vector  at 
points  equidistant  from  the  pole. 

(6)  Points  ivhere  the  locus  crosses  the  initial  line.  These  are 
found  by  putting  ^  =  0,  or  180°,  and  solving  the  resulting  equa- 
tion for  r.  If  the  equation  obtained  by  putting  ?•  =.  0  in  the  given 
equation  is  satisfied  by  some  value,  or  values,  of  0,  then  the  locus 
passes  through  the  pole. 

(c)  Limits  of  the  locus.  These  aie  determined  by  finding  the 
ranges  of  values  of  each  variable  for  which  the  other  has  real  values. 

(c?)  Change  of  one  variable  due  to  a  given  variation  of  the  other. 
It  is  important  to  determine  from  the  equation  how  increasing  or 
decreasing  either  variable  will  affect  the  other. 


75.   Example  IX.     Discuss  the  equation  r  =  cos  2  d. 

(a)  Since   ?■  =  cos  2  ^  =  cos2(^  +  180'^),   tlie  locus  is  symmetrical  with 
respect  to  the  pole. 

(&)  When  6  equals  zero  or  180"',  r  =  l.  Hence  the  locus  crosses  the 
initial  line  on  opposite  sides  of  the  pole  and  at  a  unit's  distance  from  the 
pole.  Again,  r  =  0  for 
^  =  4.5^  135-^,22-5^  or  3 15°. 
Therefore  the  locus  passes 
through  the  pole  four  times 
during  one  complete  devo- 
lution of  the  radius  vector. 

(c)  The  radius  vector  is 
real^for  every  value  of  6, 
but  since  cos  2  9  can  never 
be  greater  than  unity,  the 
locus  is  entirely  contained 
within  the  circle  whose  ra- 
dius is  1. 

(d)  As  d  varies  from 
—  45°  to  +  45°,  r  is  posi- 
tive and  the  point  (r,  0) 
describes  the  loop  to  the 
right  of  the  pole.  Between 
45°  and  135°,  cos  2  9,  and  ^^^  -g 
therefore  r,  is  negative  and 

the  point  {r,  6)  describes  the  loop  below  the  pole.     Between  135°  and  225°  r 
is  again  positive  and  the  point  (r,  9)  describes  the  loop  to  the  left  of  the  pole. 


90 


EQUATIONS  AND   THEIR  LOCI 


[Chap.  V. 


Finally,  between  225"^  and  315°,  r  is  negative  and  the  point  (r,  6)  describes 
the  loop  above  the  pole. 

The  locus  is  one  of  a  family  of  curves  known  as  "rose  curves"  from  the 
form  (Fig.  56). 

76.    Example  X.     Discuss  the  equation  r^  =  a^  cos  2  d. 

(ffl)  The  locus  is  symmetrical  with  respect  to  the  pole,  since  the  equation 
contains  only  the  second  power  of  r. 

(6)  The  locus  crosses  the  initial  line  at  the  points  for  which  r  =±a  and 
also  at  the  pole. 

(c)  The  radius  vector  can  be  real  only  when  6  has  a  value  between  —  45° 
and  +  45°  or  between  135°  and  225°.  For  all  other  positions  of  the  radius 
vector,  cos  2  0  is  negative  and  consequently  ?•  is  imaginary. 


■^ 

1 

0«^^g^^^^sj^^^dfl : 

IWW^^^^^^^^^^^^ 

^^^^^^^8  ■ 

'■^^^^^^^^^^M 

^^^^^^^^^^m 

1  ^^^^^^^^^^^^^^ 

^^^^^^B 

^^^^^^^^^^m 

1 

1 

H 

mmm 

" "  s 

F " 

H 

1 

% 

^^^^^^^M^^^M 

^^^^^^^^^^^^W I : 

■   ■  ^^^^^^^^^^^^^^^^^^W 

^^^^^^^^H 

^^^^^^^^^^^^m 

^ 

^ 

<S6%^Sw7Qi'7®i'&.'V  SSr^TTT 

T— ri^S3i\iSm\fe^5>^CW0<^^ 

Fig.  57 


(cl)  As  6  increases  from  0°  to  45°,  the  absolute  value  of  r  decreases  from 
a  to  0.  Again,  as  9  increases  from  —  45°  to  0°  the  absolute  value  of  r 
increases  from  0  to  a.  The  locus  consists,  therefore,  of-  two  loops  as  shown 
IB  Fig.  57. 


Arts.  76,  77]      TRANSFORMATION   OF   THE   AXES 


91 


EXERCISES 

1.   Discuss  the  following  equations  and  construct  the  corresponding  loci ; 
(a)  r  —  a  cos  3  6.         (6)  r  =  a  sin  3  0.  (c)  r  =  1  +  cos  6. 


(d)  r=- 


1 


(e)  r  =  1  —  cos  6. 


1  +  cos  d 

2.  Discuss  the  equations  r  =  a  cos  nO  and  r  =  a  sin  nd  for  n  an  even 
integer  ;  for  n  an  odd  integer.  What  is  the  difference  in  the  form  of  the 
curve  ? 

3.  Discuss  the  equation  r  =  a  tan  0  and  draw  the  corresponding  locus. 

4.  Change  the  equation  in  Example  X,  Art.  76,  to  rectangular  coordi- 
nates and  compare  with  Art.  54.     What  is  the  locus  ? 

62 


5.   Discuss  the  equation  r 


- ,  first  when  c  >  a  and  then  when 


a  —  c  cos  0 
c  <,a.     What  are  the  loci  ? 

6.  Discuss  the  equation   i-  =  2  a  sin  0  tan  0  and   draw  the   locus.      The 
carve  is  called  the  cissoid  of  Diodes. 

7.  Discuss  the  equation  f-  =  —  .     The  locus  is  the  lituus. 

e 

8.  Discuss  the  equation  r  =  a^.     The  locus  is  the  logarithmic  spiral. 


TRANSFORMATION   OF  COORDINATES 

77.    Transformation  of  the  axes.     The  equation  of  a  given  locus 
can  be  simplified  often  by  changing  the  axes  to  a  new  position  in 


;s 


jgF 


ra's: 


^^=h 


-i=7^=t-D- 


-mi 


m 


ah 


00: 


Fig.  58 


the  plane,  and  then  finding  the  equation  which  the  new  coordinates 
of  the  points  on  the  locus  satisfy.     The  operation  of  changing  the 


92 


EQUATIONS  AND   THEIR   LOCI 


[Chap.  V. 


axes  is  called  transformation  of  coordinates,  and  the  process  of 
finding  the  new  equation  from  the  old  is  called  transformation  of 
the  equation. 

When  the  new  axes  O'X'  and  0'  Y'  are  respectively  parallel 
to  the  old  axes  OX  and  OY  (Fig.  58),  the  transformation  is 
called  translation  of  the  axes.  Let  the  coordinates  of  the  new 
origin  0',  referred  to  the  old  axes,  be  h  and  k ;  the  coordinates  of 
any  point  P,  referred  to  the  old  axes,  be  x  and  y ;  and  the  coordi- 
nates of  P,  referred  to  the  new  axes,  be  x'  and  y'.  Then,  from 
either  of  the  positions  of  0'  shown  in  Fig.  58  (cf.  Art.  3),  we 

^^^^®  0D=  OE  +  ED     and     DP  =  DD'  +  D'P 

=  h  +  x'  =Jc  +  y'. 

Hence,  x  =  h  -\-  x'  and         y  =  k  -\-  y'.  (1) 

It  is  clear  that  equations  (1)  hold  wherever  the  point  0'  may 
be  situated,  provided  the  new  axes  have  the  same  direction  as  the 
old. 

Example.  Transform  the  equation  y  =  2x  +3  by  translating  the  axes 
so  that  the  new  origin  shall  be  the  point  (1,  5). 

Here,  for  any  point  (x,  «/)  in  the  plane, 
and  hence  for  any  point  on  the  locus  of 
the  given  equation, 

x=  1  +  x'     and    y  =  5  +  y'. 

Therefore,  in  terms  of  x'  and  y',  the 
given  equation  becomes 

5  -f-  ?/'  =  2(1  +  x'),  or  y'  =  2x'.     (2) 

The    given   equation   y  =  2  x  +  3 
and  the  new  equation  y'  =  2  x'  repre- 
sent the    same   locus  ;   namely,  the 
straight  line  shown  in  Fig.  59.    The 
origin  is,  in  the  one  case,  at  0,  and  in  the  other,  at  0'. 

78.  Rotation  of  the  axes.  When  the  origin  is  not  moved,  but 
the  axes  are  each  rotated  through  a  given  angle,  the  transforma- 
tion is  called  rotation  of  the  axes. 

To  obtain  the  equations  for  rotating  the  axes,  let  P  be  any 
point  in  the  plane  (Fig.  60)  whose  coordinates  referred  to  the  old 


Fig.  59 


Art.  78] 


ROTATION   OF   THE   AXES 


93 


axes  are  (x,  y),  and  referred  to  the  new  axes  are  (x',  y').     Let  the 
angle  XOX',  through  which  the  axes  are  rotated,  be  denoted  by 


Fig.  60 


e,  the  angle  XOP  by  <^,  and  the  angle  X'OP  by  «.     If  OP  =  r, 

then 

X  —  r  cos  <^  =  r  cos  (d  -\-  a)  =  r  cos  9  cos  a  —  r  sin  6  sin  a, 
y  =  r  sin  <^  =  r  sin  (^  +  «)  =  r  sin  9  cos  «  +  r  cos  ^  sin  «. 


But  r  cos  a  —  x'  and  r  sin  «  =  ?/'  ;  and  therefore 

x  =  oc'  cos  6—2/'  sill  6, 
y  =  x'  sin  8  +  z/'  cos  0. 


(1) 


These  equations  express  the  old  coordinates  of  any  point  in 

terms  of  the  new  coordinates.     To  obtain  the  new  coordinates 

in  terms  of  the  old,  we  can  solve  equations  (1)  for  x'  and  y',  or 

we  can  derive  x'  and  y'  directly  from  the  figure.     In  either  way 

we  find  ,  /I  ■     /, 

X  =  X  cos  9  +  y  sm  9, 

y'  —  y  cos  9  —  X  sin  9. 


(2) 


Example.     Transform  the  equation  24  xy  —  7  y^  =  144  by  rotating  the 
axes  through  the  acute  angle  whose  tangent  is  f . 


94 


EQUATIONS  AND  THEIR  LOCI 


[Chap.  V. 


Here  sin  ^  =  |  and  cos  d=  f ,  hence  the  equations  for  rotating  the  axes  are 


X  =  f  X'  — 


y'. 


Substituting  in  the  given  equation  and  reducing,  we  have 

9  a;'2  _  16  m'2  =  144,  or  —  -  ^^  =  1. 
^  1(3       9 

The  given  equation,  therefore,  represents  an  liyperbola  whose  semiaxes  are 
4  and  3. 

EXERCISES 

1.  Transform  the  equation  3  x  +  7  y  =  8  to  a  new  set  of  axes  parallel  to 
the  old  set,  and  having  the  point  (4,  —  2)  as  origin. 

2.  Show  that  the  equation  x^  +  y'^  =  a^  ig  unaltered  by  rotating  the  axes 
through  any  angle  0.    What  is  the  geometrical  interpretation  of  this  fact  ? 

3.  Transform  the  equation  x'^  —  y^  =  10  by  rotating  the  axes  through  an 
angle  of  45°. 

4.  Transform   the  equation  x^  +  y^  =  a^   by  first  translating  the  axes 
parallel  to  themselves,  the  new  origin  being  at  the  point  ( - ,  -  | ,  and  then 

rotating  the  new  axes  through  the  angle  45°.     What  is  the  locus  of  the 
resulting  equation  ?     What  is  the  locus  of  the  original  equation  ? 

79.    Removal  of  terms  of  first  degree.     When  the  given  equation 
is  an  algebraic  equation  of  the  second  degree  in  the  variables  and 

contains  terms  of  the 
first  degree,  the  latter 
can  often  be  removed 
by  translating  the  axes 
to  a  new  origin,  as  il- 
lustrated by  the  follow- 
ing example. 

Example.  Given  the 
equation  4  x^  +  9  ?/'-  —  16  x 
—  18  y  =  11.  Translate  the 
axes  so  as  to  remove  the 
terms  of  first  degree. 

Substituting  for  x  and 
y    their    values    in    terms 
of  x'  and  ?/',  equations  (1),  Art.   77,  we  have 

4(x'  +  ^0^  +  9(2/'+  *)^  -  1<K^'  +  '0  -  18(2/'  +  ^0  =  11- 


%<^r. 


?^: 


# 


-X- 


m 


Fig.  61 


Arts.  79,  80]     REMOVAL   OF  THE   TERM   IN   xy  95 

The  coefiBcients  of  x'  and  y'  in  this  new  equation  are  respectively  8  /i  —  16 
and  18  ^•  —  18.  Hence,  if  we  choose  h=2  and  k  =  \,  the  terms  of  first  degree 
will  drop  out  of  the  new  equation  and  it  reduces  to 

4  x>'^  +  9  ?/'-  =  36. 

This  equation  represents  an  ellipse  whose  semiaxes  are  3  and  2,  hence  the 
given  equation  represents  this  ellipse.  Figure  61  shows  the  curve  and  both 
sets  of  coordinate  axes. 

80.  Removal  of  the  term  in  ocij.  The  term  in  xy  can  be  removed 
from  an  equation  of  the  second  degree  by  rotating  the  axes 
through  the  proper  angle.     This  is  illustrated  in  Art.  78. 

As  another  example,  we  will  remove  the  term  in  xy  from  the  equation 

x2  -f  2  xy  +  2  2/2  -  4  =  0. 

Substituting  the  values  of  x  and  y  from  equations  (1),  Art.  78,  and  collecting 
terms,  the  given  equation  becomes 

(cos2  ^  +  2  sin2  ^  +  2  cos  6*  sin  e)x'-  +  (2  cos  6  sin  d  +  2  cos^  0  —  2  sin^  e)x'y'  + 
(sin2  Q  ji^2  cos'-  e  —  2  sin  6  cos  e)y''^  —4=0.  (1) 

Putting  the  coefficient  of  x'y'  equal  to  zero,  we  have  the  equation 

2  cos  d  sin  0  +  2  (cos2  6  —  sia?  6)  =  0, 
from  which  to  determine  6.     But  this  equation  is  equivalent  to 

sin  2  ^  +  2  cos  2  ^  =  0  or  tan  2  ^  =  —  2. 
Therefore  2  6*  =  arc  tan  (—  2)  =  116"'  34',  nearly,  or  ^  =  58°  17',  nearly. 
Since  tan  2  fi"  =  -  2, 

2  —  1 

we  have  sin  2  ^  = ,  cos  2  ^  =  - 


Hence,  cos^  d 


V5  V5 

1  +  cos 2 g _  \/.5—  1^ 
2  ~    2V5 

1  —  cos  2  g  _  V'5  +  1  _ 

V2  2\/5 

sin  2  ^        1 


and  sin  6  cos  9  = 

2  V5 

Substituting  these  values  in  (1),  it  reduces  to 


3  _  V5      3  +  V5 
Hence  the  given  equation  represents  an  ellipse. 


96  EQUATIONS  AND  THEIR   LOCI  [Chap.  V. 

EXERCISES 

1.  Remove  the  terms  of  first  degree  and  then  the  term  in  xy  from  the 
equations  ^^^  xy-x-y  =  0;  (ft)  xy-x  +  y  =  0. 

What  are  the  loci  which  these  equations  represent  ? 

2.  Show  that  the  terms  of  first  degree  cannot  be  removed  from  the 
equation  ^g  ^^  -  24:xy  +  9y^  -  20x- 110  y  +  225  =  0. 

Try  to  generalize  this  result  so  as  to  tell  at  a  glance  whether  the  terms  of 
first  degree  can  be  removed  or  not  from  any  equation  of  the  second  degree. 

3.  Given  the  equation  x^  —  3  axy  +  y^  =  0.  Rotate  the  axes  through  the 
angle  45°  and  compare  the  resulting  equation  with  Example  V,  Art.  70. 
What  locus  does  the  given  equation  represent  ? 

4.  In  exercise  2,  rotate  the  axes  through  the  angle  6  =  arc  tan  |  and 
then  translate  the  axes,  taking  for  new  origin  the  point  (2,  1).  What  locus 
does  the  equation  represent  ? 

81.    Classification  of  algebraic  curves. 

Theorem.  The  degree  of  an  algebraic  equation  in  the  variables  x 
and  y  is  unaltered  by  transformation  of  coordinates. 

For,  in  transforming  the  equation  by  translation  or  rotation  of 
the  axes  we  replace  x  and  y  by  expressions  of  the  first  degree  in 
x'  and  y'  and  therefore  the  degree  of  the  equation  cannot  be  raised 
by  this  process.  Neither  can  it  be  lowered,  for  then  it  would  be 
necessary  to  raise  the  degree  in  transforming  back  to  the  original 
axes,  and  Ave  have  just  seen  that  the  degree  cannot  be  raised  by  a 
transformation  of  coordinates.  Since  the  degree  cannot  be  raised 
or  lowered  by  a  transformation  of  coordinates,  it  must  remain 
unaltered. 

Because  of  the  theorem  just  proved,  algebraic  curves  can  be 
classified  conveniently  according  to  the  degree  of  their  equations, 
since  we  now  know  that  the  degree  of  the  equation  is  independent 
of  the  position  of  the  axes  with  reference  to  the  curve. 

If  the  degree  of  a  given  algebraic  equation  is  any  integer  n, 
the  corresponding  curve  is  said  to  be  of  order  n.  The  straight 
line  is  the  only  locus  of  order  1  (Art.  47).  The  circle,  the  ellipse, 
the  hyperbola,  and  the  parabola  are  all  loci  of  order  2.  The 
folium  of  Descartes  is  a  locus  of  order  3  (Art.  70).  The  ovals  of 
Cassini  are  loci  of  order  4  (Art.  54). 


Art.  81]     CLASSIFICATION   OF  ALGEBRAIC   CURVES  97 

The  succeeding  chapters  will  be  devoted  to  a  special  study  of 
loci  of  orders  1  and  2. 

MISCELLANEOUS    EXERCISES 

1.  Show  that  the  triangle  whose  vertices  are  (3,  2),  (—1,  —3),  and 
(—  6,  1)  is  a  right  triangle. 

2.  On  the  line  ?/  —  5  =  0  a  segment  is  laid  off,  having  for  abscissas  of  its 
extremities  2  and  5,  and  upon  this  segment  an  equilateral  triangle  is  con- 
structed.    What  are  the  coordinates  of  its  third  vertex  '? 

3.  Find  the  coordinates  of  the  point  dividing  the  segment  (5,  2)  to 
(4^  _  7)  in  the  ratio  4  :  7. 

4.  Change  the  polar  equations  =  a  -| to  one  in  rectangular  coordi- 

cos^ 

nates.     Plot  the  locus. 

5.  Remove  the  terms  of  first  degree  from  the  equation4a;--|-9  j/  — 8  y— 6=0 
and  plot  the  resulting  equation. 

6.  Find  the  rectangular  equations  of  the  asymptotes  of  the  hyperbola 
whose  polar  equation  is 

~  4  cos  61  —  3 

7.  Discuss  the  equation  y-  =  4:x:^  —  x^  and  plot  the  locus.  Write  the 
parametric  equations  of  the  locus  if  y  =  tx.  t  being  the  parameter. 

2  4-  s^ 

8.  Discuss  the  equation  y-  =:x-  and  plot  the  locus.    Write  the  para- 

2  —  X 
metric  equations  if  y  =  tx. 

Ill 

9.  Write  the  parametric  equations  of  the  locus  ofx-  +  y^  =  a'^,  assuming 

X  —  a  COS'*  d. 

10.  OB  is  the  crank  of  an  engine  and  AB  the  connecting  rod,  A  being 
the  piston.  A  moves  in  a  straight  line  passing  through  0.  Find  the  equa- 
tion of  the  locus  of  any  point  P  on  the  connecting  rod.  Let  P  be  a  units 
from  A  and  6  units  from  B,  and  let  r  be  the  length  of  the  crank,  OB.  Dis- 
cuss the  equation  and  plot  the  locus.  Write  the  parametric  equations  of  the 
locus,  assuming  y=  a  sin  d.     What  is  the  locus  when  r  =  a  +  b? 

11.  Discuss  the  equation  y-(a  —  x)— oc^(a  +  x)=  0,  and  plot  the  locus. 
The  curve  is  called  the  strophoid. 

12.  Construct    the  locus    of     w  =  sin  2  x  H ;     of    ?/ =  e-^  +  4  x'-^ :    of 

y  =  -  ■  cos  X. 

X 

13.  Discuss  the  equation  r^  =  a^  sin  2  6  and  plot  the  locus. 

14.  Discuss  the  equation  r^  =  o?-  tan  Q  and  plot  the  locus. 


CHAPTER   YI 
LOCI    OF   FIRST   ORDER 

82.  Linear  equations.  We  have  seen  (Art.  47)  that  the  equa- 
tion of  every  straight  line  is  of  the  first  degree  in  x  and  y ;  and 
conversely,  that  every  equation  of  first  degree  in  x  and  y  is  the 
equation  of  a  straight  line.  For  this  reason,  an  equation  of  the 
first  degree  in  x  and  y  is  called  a  linear  equation. 

Every  linear  equation  is  of  the  form 

Ax  +  5?/  +  C  =  0,  (1) 

where  A,  B,  and  C  are  constants.  These  constants  can  have  any 
values,  except  that  A  and  B  cannot  both  be  zero,  for  then  the  equa- 
tion would  contain  neither  variable.    If  A  is  zero,  (1)  is  the  equation 

C 
of  a  line  parallel  to  the  X-axis,  for  then  y  has  the  value for 

all  values  of  x.     If  B  is  zero,  (1)  is  the  equation  of  a  line  parallel 

C 

to  the  F-axis,  for  then  x  has  the  value for  all  values  of  y. 

A  -^ 

Finally,  if  C  is  zero,  (1)  is  the  equation  of  a  line  passing  through 

the  origin,  for  then  the  equation  is  satisfied  when  x  =  0  and  y  =  0. 

For  A,  B,  C  different  from  zero,  the  slope  of  the  line  is  given 

by  the  formula  j 

and  the  intercepts  a,  b  on  the  X-  and  I'-axes  by 

«=  — -T   and   ^—~^' 

respectively. 

EXERCISES 

1.  Find  the  slopes  and  intercepts  of  the  lines  whose  equations  are  the 
following : 

(a)  x  +  VSy  +  10  =  0.  {b)y  =  x-6. 

(c)   5  X  -  12  2/  =  13.  (d)  2  X  -  3  y  =  4. 

98 


Arts.  82,  83]      INTERSECTION   OF  TWO   LINES  99 

(e)  a;  -  rt  =  0.  (/)  4  ?/  +  3  x  =  24. 

(g)  5x  +  4:y=20.  (h)  2x  ~  iy  +  9  =0. 

(i)    2x  +  3y=0.  ij)  2/ =4. 

(k)  Ax  +  By+  C  =  0.  (I)  (a^  -  b-^)x  =(a  + b)y +  c. 

2.    If  a  and  6  represent  the  intercepts  on  the  J"-  and  I^'-axes  respectively, 
and  m  the  slope,  determine  the  equations  of  the  lines  for  which 
(l)a  =  2,  &=-3.  (2)a=-l.m  =  4. 

(3)  6  =3,  wi  =  -2.  (4)  m=-5,  a=-2. 

(5)  a  =2,  and  passing  through  the  point  (4,  —  3). 

(6)  Passing  through  the  points  (—  1,  2)  and  (5,  —  4). 

^  ^  C  C 

83.  Intersection  of  two  lines.  The  coordinates  of  the  point  of 
intersection  of  two  lines  must  satisfy  the  equation  of  each  line, 
since  the  point  lies  on  each  line.  To  find  the  coordinates  of  the 
point  of  intersection  it  is  only  necessary,  therefore,  to  solve  the 
equations  simultaneously  for  x  and  y.  For  example,  x  =  3  and 
2/  =  4  is  the  common  solution  of  the  two  equations  x  —  y  -\-  1  =  0 
and  Ax  +  y  —  16  =  0;  and  these  are  the  equations  of  two  straight 
lines  which  intersect  in  the  point  (3,  4). 

In  general,  two  straight  lines  intersect  in  one  and  only  one 
point.     But  they  may  be  : 

(1)  Parallel  to  each  other. 

(2)  Coincident, 

In  the  first  case  the  slopes  of  the  lines  are  equal  and  their 
equations  have  no  common  solution.     For  example,  the  equations 

2  a;  -  3  !/  =  4  and  2  a-  -  3y  =  7  (1) 

are  the  equations  of  a  pair  of  parallel  lines,  since  the  slope  of 
each  is  -|.  The  equations  have  no  common  solution.  The  equa- 
tions of  a  pair  of  parallel  lines  are  called  incompatible  or  inconsist- 
ent* Thus  equations  (1)  are  incompatible.  Obviously  2 ;i;— 3 2/ 
cannot  be  4  and  7  at  the  same  time  for  any  values  of  x  and  y. 

In  the  second  case  the  slopes  of  the  lines  are  also  equal,  but 
their  equations  have  an  indefinite  number  of  common  solutions, 
since  any  pair  of  values  of  x  and  y  that  satisfies  one  equation  must 
also  satisfy  the  other.     The  equations,  therefore,  can  differ  only 

*  See  Rietz  and  Crathorne,  College  Algebra,  p.  49. 


100  LOCI   OF  FIRST   ORDER  [Chap.  VI. 

by  a  constaut  factor.  The  equations  of  a  pair  of  coincident  lines 
are  called  dependent.     Thus,  the  equations 

'2x  —  3y  =  4:  and  4 a;  —  6 ?/  =  8 
are  dependent  and  are  the  equations  of  a  pair  of  coincident  lines. 

EXERCISES 

1.  Find  the  intersections  of  the  lines  represented  by  the  following  pairs 
of  equations.    Tell  which  are  inconsistent  and  which  are  dependent  equations, 
(a)  2x  + 3?/  =  12,  4x- ?/  =  5.  (6)  3x  +  5  ?/ =  1,  6a;  +  10?/ +  7  =  0. 
(c)  5  X  -  2  ?/  +  4  =  0,  a;  -  .4  ?/  =  -  .8.    (d)  x  +  3  ?/  =  15,  3  x  -  ?/  =  5. 
Draw  the  lines  in  each  case. 

2.  Write  an  equation  representing  the  same  straight  line  as  5x+4i/— 20=0 
in  which  the  sum  of  the  coefScients  shall  be  22 ;  in  which  the  product  of  the 
first  and  third  coefficients  shall  be  equal  to  the  second. 

3.  Change  the  equation  3x  —  4?/  +  12  =  0  into  another  representing  the 
same  straight  hne  and  having  the  square  of  the  second  coefficient  equal  to 
twice  the  third  minus  four  times  the  first ;  having  the  product  of  all  three 
coefficients  equal  to  minus  three  times  the  last. 

4.  The  equations  5x  —  2?/  —  3  =  0  and  Ax  +  By  +  C  =  0  are  dependent 
and  B^-\-2(A  +  C)  =  24.     Find  A,  B,  and  C. 

84.  The  pencil  of  lines.  Let  m  =  0  and  v  =  0  he  the  equations  of 
two  straight  lines,  then  the  equation 

u  +  fcv  =  0  (1) 

is  the  equatio7i  of  a  straight  line  passing  through  the  intersection  of 
ic  =  0  and  v  =  0,  ivhatever  value  is  given  to  Tc. 

Here  u  and  -v  are  each  expressions  of  the  first  degree  in  x  and 
y  and  therefore  u  -{■  hv  =  0  is  the  equation  of  some  straight  line 
(Art.  47).  Moreover,  u  +  A;v  =  0  is  satisfied  by  the  coordinates 
of  the  point  of  intersection  of  u  =  0  and  v  =  0  and  therefore  it  is 
the  equation  of  a  straight  line  passing  through  the  intersection 
of  u  =  0  and  v  =  0. 

If  k  is  allowed  to  vary,  a  series  of  lines  is  obtained  each  pass- 
ing through  the  intersection  of  m  =  0  and  v  =  0.  The  totality  of 
lines  so  obtained  is  called  a  pencil  of  lines  (Fig.  62). 

The  constant  k  can  be  determined  so  that  the  line  u  -\-  kv  =  0 
shall  satisfy  any  single  condition,  such  as  passing  through  a  given 
point,  having  a  given  slope,  etc. 


Art.  84] 


THE   PENCIL   OF  LINES 


101 


Example.  Find  the  equation  of  the 
line  passing  through  the  intersection  of 
2x  +  3y  -  4:  =  0  and  x  +  2?/— 5  =  0 
and  also  through  the  point  (2,  3). 

The  line  whose  equation  is  required 
is  one  of  the  pencil 
2x  +  Sy  -4:  +  k{x  +  2y-5)=0.    (2) 

Since  tlie  line  is  to  pass  through  the 
point  (2,  3),  these  coordinates  must 
satisfy  the  equation.  Hence  k=—3. 
Substituting  this  value  of  k  in  (2),  we 
have  the  required  equation, 
x  +  Sy-ll  =0. 

This  result  may  be  verified  by  solving 
the  given  equations  simultaneously  and 
then  finding  the  equation  of  the  line 
passing  through  the  common  point  and 
the  point  (2,  3)  in  the  usual  way.* 


Fig.  62 


EXERCISES 

1.  What  is  the  equation  of  the  line  passing  through  the  origin  and  through 
the  intersection  of  the  lines  x  +  Sy  —  8  =  0  and  4 x  —  5 ?/  =  10  ? 

2.  The  equations  of  the  sides,  of  a  triangle  are  5  a;— 6?/ =  16,  4x  +  5  2/  =  20, 
and  z  +  2y  =  0.  Find  the  equations  of  the  lines  passing  through  the  ver- 
tices and  parallel  to  the  opposite  sides. 

3.  Find  tlie  equation  of  the  line  which  passes  through  the  intersection  of 
the  lines  2  x  —  3  y  +  1  =  0  and  a;  +  5  ?/  +  6  =  0  and  is  perpendicular  to  the 
first  of  these  lines.     Which  is  parallel  to  the  line  6x  —  y  +  10  =0. 

4.  What  is  the  equation  of  the  line  which  passes  through  the  intersection 
of  the  lines  ?/  =  7  x  —  4  and  y  =—  2 x  +  5  and  makes  an  angle  of  60°  with  the 
positive  end  of  the  x-axis  ? 

5.  Find  the  equation  of  the  line  which  passes  through  the  intersection  of 
the  lines  by  —  2x—  10  =  0  and  ?/-!-4x  —  3  =  0,  and  also  through  the  inter- 
section of  the  lines  10  2/-fx-f-2l  =  0  and  3y  —  6x  +  1  =  0. 

Suggestion.     The  equations 
6y  —  2x-  10  +  k{y  +  4x  -  3)  =  0  and  10?/ -1-  a; -|- 21  -|-  k'(3y  —  5x  +  1)  =  0 
must  be  dependent  (Art.  83). 

*  The  theorem  of  this  article  holds  when  u  and  v  are  expressions  of  any  degree 
in  X  and  y.    2(  +  kv  =0  is  then  the  equation  of  a  pencil  of  curves. 


102  LOCI   OF  FIRST   ORDER  [Chap.  VI. 

85.  The  pair  of  lines.  Let  u  =  0  and  v  =  0  be  the  equations  of 
two  straight  lines,  then  the  locus  of  the  equation 

is  the  jKtir  of  lines  m  =  0  and  v  =  0  taken  together. 

For  the  equation  u  ■  v  =  0  is  satisfied  only  when  one  of  the 
factors,  u  or  v,  is  equal  to  zero  or  when  both  factors  are  equal  to 
zero.  But  the  two  straight  lines  pass  through  all  the  points 
whose  coordinates  satisfy  the  equations  u  =  0  and  v  =  0,  and 
through  no  others.  Consequently  these  lines,  taken  together, 
form  the  locus  of  the  equation  u  •  v—0  (Art.  63).* 

Example.  The  locus  of  the  equation  x-  —  y'^  =  0  is  the  pair  of  lines 
X  +  ?/  =  0  and  x  —  y  =  0. 

EXERCISES 

1.  Draw  the  pairs  of  lines  whose  equations  are  the  following  : 

(a)  x^  +  xy  =  Q.  (b)  2x^  +  5xy  -  3y^  =  0.  (c)  x^  —  5x  +  Q  =  0. 

{d}  2  if  -  xy  +  4:X  -  9  y  +  i  =  0.         (e)  x'^  -  ?/  -  2  ?/  -  1  =  0. 

2.  Write  the  equation  of  the  pair  of  lines  each  of  which  passes  through 
the  origin  and  whose  slopes  are  respectively  V3  and  —  VS. 

3.  Write  the  equation  of  the  pair  of  lines  each  of  which  passes  through 
the  point  (1,  2)  and  whose  slopes  are  respectively  2  and  —  |. 

86.  The  normal  form.  We  have  seen  (Art.  57)  that  the  polar 
equation  of  a  straight  line  is 

r  cos  (6  —  a)  =p, 

where  j9  is  the  length  of  the  perpendicular  from  the  origin  on 
the  line  and  «  is  the  inclination  of  this  perpendicular  to  the 
X-axis. 

Expanding  cos  {9  —  a),  the  equation  becomes 

r  (cos  6  cos  a  +  sin  6  sin  a)=p. 

Since   r  cos  6  =  x  and   r  sin  6  =  y,    the    equation    in   rectangular 

coordinates  is  .  ... 

X  cos  a  +  1/  sma  =  p.  ■  (1) 

*  The  theorem  of  this  article  holds  when  %(■  and  r  are  expressions  of  any  degree 
in  X  and  y.    The  locus  oi  u-  v  =  0  is  then  called  a  composite  curve. 


Arts.  85-87]        REDUCTION  OF  Ax  +  By  +  C  =  0 


103 


This  is  called  the  normal  form  of 
the  equation  of  a  straight  line  (Fig. 
63). 

87.    Reduction  of  Ajc+Bi/  +  C=0 

to  the  normal  form.  The  problem 
before  us  consists  in  reducing  the 
given  equation 

Ax  +  Bi/+C  =  0  (1) 

to  the  form 

xcosa  +  y  sina—p  =  0.      (2) 

Since  (2)  is  to  be  the  equation  of  the  same  line  as  (1),  the  two 
equations  can  differ  only  by  a  constant  factor  (Art.  83).  Hence 
we  must  have 

cos  a  =  kA,  sin  «  =  JcB,  and  —  2>  =  kC,  (3) 

where  Jc  is  the  constant  factor.  F.rom  the  first  two  of  these  equa- 
tions, by  squaring  and  adding,  Ave  get 


Fig.  (i3 


l  =  k^(A-'  +  B'-),  or  k  = 


Therefore 


±  VA^  +  £2 


cos  a  = 


,  SlU  a  = 


B 


:,  and  —  p 


C 


.  .     (4) 

In  order  to  determine  which  sign  shall  be  given  to  the  radical 
in  any  numerical  example,  we  shall  assume  that  |>  is  always  a 

positive  number  and  then,  from 
(4),  the  sign  of  the  radical  must 
he  opposite  to  the  sign  of  C. 

For  example,  to  reduce 
3a;+4?/  +  10  =  0  to  the  normal 
form,  we  divide  both  members 
of  the  equation  by  —  V9  + 1" 
=  —  5  and  obtain 

Fig.  6i  Therefore 

p  =  2,  cos  a  =  —  1^,  sin  «  =  —  A,  and  a  =  233°  8' 
(Fig.  64). 


104 


LOCI   OF  FIRST   ORDER 


[Chap.  VI. 


EXERCISES 

1.    Write  the  equations  of  the  lines  for  which  ; 


(«)  p  =  5,  a  =  60°. 
(d)  p  =  0,  a  =  225°. 


(6)  p  =  5,  a  =  120°. 
(e)  p  =  l,  a  =  45°. 


(c)  p=-6,  a  =  330°. 
(/)  p  =  6,  a  =-60°. 


2.  Reduce  the  following  equations  to  the  normal  form  and  plot  the  lines 
of  which  they  are  the  equations  : 

(a)  4x-3y  -25.  (b)  :«•  +  4  =  0.  {c)  x  +  2ij  --  8. 

•      (fZ)52/-3=0.  {e)2x-y-0.  (/)  x  -  3?/ +  4  =  0. 

3.  What  system  of  lines  is  given  by  xcos  a  +  y  sin  a —p  =  0  when  a 
is  constant  and  p  varies  ?     When  p  is  constant  and  a  varies  ? 

4.  Two  lines  can  be  drawn  through  the  point  (2,  5)  and  tangent  to  the 
circle  x'^  +  y^  =  25.     Find  the  equation  of  each  line.     Draw  the  figure. 

88.   Distance  from  a  line  to  a  point.     With   the   help   of  the 
normal  form  of  the  equation  of  a  straight  line,  it  is  easy  to  find 

the  length  of  the  perpen- 
dicular DP  drawn  from  a 
given  line  AB  to  a  given 
point  P(x„  y,)  (Fig.  65). 
Thus,  let  the  equation  of 
AB,  reduced  to  the  normal 
form,  be 

yS    a^cos  «  +  ?/sin  a  —  ^:>  =0.  (1) 

Through  P  draw  BS  paral- 
lel to  AB,  and  let  0M  =  jh 
be  the  length  of  the  per- 
pendicular from  0  to  ES. 
Then   the  normal   form    of   the    equation   of   PS   is 


^(^vVi) 


Fig.  65 


X  COS  u-\-y  sin  «  —  2h  —  0- 


(2) 


Since  P  is  on  RS,  the  coordinates  Xj^  and  y^  must   satisfy  (2). 

Hence,  . 

Xi  cos  « -f  ?/i  sm  «  =  2h-  (<j) 

Subtracting  p  from  both  members  of  (3),  we  have 

a'l  cos  a  +  ?/i  sin  a  —  'p=lh  —P=  OM—  0K=  DP. 


Arts.  88,  89]    DISTANCE  FROM  A  LINE  TO  A  POINT     105 


Hence  the  following  rule : 

To  find  the  distance  from  a  line  to  a  point,  reduce  the  equation 
of  the  given  line  to  the  normal  form  with  the  right  member  equal  to 
zero;  substitute  the  coordinates  of  the  given  point  in  the  left  member. 
The  result  is  the  required  distance. 

The  sign  of  the  result  will  be  negative  when  p^  <  p ;  that  is, 
when  the  given  point  and  the  origin  are  on  the  same  side  of  the 
given  line.  The  sign  will  be  posi- 
tive in  the  contrary  case. 

Example.     Find  the  distance  from  tlie 
line  3  X  +  i  ?/  +  12  =  0  to  the  point  (2,  3). 
Here   the   normal   form   of   the   given 
equation  is 

Sx  +  4:y +  12  ^Q 
-5 

Substituting  2  and  3  for  x  and  y  in  the 
left  member,  we  find  the  required  distance 
is  —  6.     Figure  66  illustrates  the  example, 
the  origin  are  on  the  same  side  of  the  line. 


Note  that  the  given  point  and 


EXERCISES 

1.  Find  the  distance  of  the  point  (3,  5)  from  the  line  2x— 3?/  +  6  =  0. 

2.  Find  the  distance  between  the  parallel  lines  7  x  —  8  y  —  15  and 
7  .r  -  8  2/  =  40. 

3.  The  line  5x  +  l2y  =  25  touches  a  circle  whose  center  is  the  origin. 
Find  the  radius  of  the  circle  and  write  its  equation. 

4.  The  line  6x  — 8?/  =  15  touches  a  circle  whose  center  is  the  point 
(—  3,  4).     Find  the  radius  of  the  circle  and  write  its  equation. 

5.  Find  the  equations  of  the  circles  inscribed  in  the  following  triangles :  — 

(a)  x  +  2y  —  5  =  0,  2  x  —  y  —  5  =  0,  2x  +  y  +  5  =  0. 

(6)  3x  +  y-l  =  0,  x-3y-S  =  0,  x  +  3y  +  11  =  0. 

(f)  X  +  2  =  0,  y-S  =  0,  X  +  2/  =  0. 

(d)  x  =  0,  y  =  0,  X  +  y  +  o  =  0. 

89.  The  angle  which  one  line  makes  with  another.  When  the 
equations  of  two  lines  are  given,  the  slopes  of  the  lines  are  known. 
The  tangent  of  the  angle  which  one  line  makes  with  the 
other  can  then  be  computed  as  in  Art.  13.  For  example,  we 
will  find  the  angle  which  the  line  x  +  2y  —  3  =  0  makes  with  the 


106  LOCI   OF  FIRST   ORDER  [Chap.  VI. 


line  Sec— ?/  +  4  =  0.  Here  the  slope  of  the  first  line  is  711^  =  —  ^ 
and  the  slope  of  the  second  line  is  Wa  =  3.  Therefore,  by  formula 
(4),  Art.  13,  we  have 

tan  <^  =  — =  — '—^  —  —  7. 

1  +  wijmj      1  —  I 

The  angle  <^  is  approximately  98°  8'.  The  student  should  con- 
struct the  figure  to  illustrate  this  example. 


EXERCISES 

1.  Find  the  angle  which  the  line  3x  —  y-}-2=0  naakes  with  the  line 
2x  +  y-2  =  0. 

2.  Find  the  angle  which  the  line  2x  —  By  +  1  —  0  makes  with  the  line 
x-2?/  +  3=0. 

3.  Find  the  angles  of  the  triangle  formed  by  the  lines  x  +  oy  —4t  =  0, 
Sx-2y  +  l=0,  and  x  —  y  +  S  =  0.     Draw  the  figure. 

4.  Find  the  equations  of  the  bisectors  of  the  angles  formed  by  the  two 
lines  3x  —  4?/  +  2  =  0  and  Ax  —  Sy  —  1  =  0.  Show  that  the  bisectors  are 
perpendicular  to  each  other. 

Suggestion.  Any  point  on  the  bisector  of  an  angle  is  equidistant  from 
the  lines  forming  the  angle. 

5.  Find  the  equations  of  the  bisectors  of  the  interior  angles  of  the  tri- 
angles formed  by  the  lines  5  x  -  12  y  =  0;  5  x  +  12  ?/  -|-  60  =  0,  and  12  x  —  by 
—  60  =  0.     Show  that  the  bisectors  meet  in  a  point. 

6.  Generalize  the  preceding  exercise  and  thus  show  that  the  bisectors  of 
the  interior  angles  of  any  triangle  meet  in  a  point.  Choose  the  coordinate 
axes  so  that  the  equations  of  the  sides  of  the  triangle  are  as  simple  as  possible. 

7.  Show  that,  for  any  straight  line, 

P      ■  P         J  ^  1 

cos  a  =  — ,  sma  =~,  and  tan  a  = , 

a  0  m 

where  a  and  b  are  respectively  the  X-  and  Y-intercepts,  m  is  the  slope,  p  is 
the  perpendicular  from  the  origin  on  the  line,  and  a  is  the  inclination  of  this 
perpendicular  to  the  X-axis. 

8.  Write  the  equation  of  tlie  straight  line  for  which  : 

(l)a  =  3,  6=-2;     (2)ffl  =  5,  _p  =  3;     (3)  m  =  f ,  p  =  5. 

9.  Of  the  five  numbers  a,  b,  in,j),  and  a,  having  given  any  two,  the  other 
three  can,  in  general,  be  determined.  What  cases  form  an  exception  to  this 
general  rule  ?    From  the  given  pairs  in  exercise  8,  determine  the  other  three. 


CHAPTER   VII 
LOCI  OF  SECOND  ORDER.    EQUATIONS  IN  STANDARD  FORM 

DIRECTRICES 

90.  Review.  We  have  found  that  the  circle,  the  ellipse,  the 
hyperbola,  and  the  parabola  are  curves  of  the  second  order  (Art. 
81).  The  definitions  of  these  curves  and  also  the  process  of  find- 
ing the  standard  forms  of  their  equations  should  be  reviewed 
(Chapter  IV). 

In  this  chapter  we  shall  derive  some  important  properties  of 
these  curves,  making  use  of  the  standard  forms  of  the  equations. 

EXERCISES 

1.  If  a  and  h  represent  the  lengths  of  the  semiaxes  and  e  the  eccentricity 
write  the  standard  equation  of  the  ellipse  for  which  ;  — 

(1)  «  =  3  and  6  =  2.  (4)  6  =  4  and  c  =  ae  =  3. 

(2)  6  =  3  and  e  =  \.  (5)  a  =  5  and  c  =  3. 

(3)  a  =  Q  and  e  =  f .  (6)   c  =  4  and  e  =  J. 

2.  Find  a,  6,  c,  and  e  from  the  following  equations  :  — 

(1)  a;2  -  25  2/2  =  25.  (4)  9  x2  +  4  2/2  =  36. 

(2)  x2  +  25  2/2  =  25.  (5)  2x^  -bif  =  20. 

(3)  9  a;2  -  4  ^2  =  36.  (q-^  2  ic2  4-  5  2/2  =  20. 

3.  Find  the  lengths  of  the  focal  radii  of  the  ellipse  x^  -\-Qy'^  =  18,  drawn 
to  the  points  whose  abscissa  is  —  2. 

4.  Find  the  lengths  of  the  focal  radii  of  the  hyperbola  9x2  —  4  2/2  =  65, 
drawn  to  the  points  whose  ordinate  is  2. 

5.  Show  that  the  circle  is  the  limiting  form  of  the  ellipse  as  a  and  6 
approach  equality.  What  is  the  eccentricity  of  the  circle  and  where  are  its 
foci? 

6.  When  the  semiaxes  of  an  hyperbola  are  equal,  the  hyperbola  is  called 
equilateral.  Show  that  the  distance  from  any  point  on  an  equilateral  hyper- 
bola to  the  center  of  the  curve  is  a  mean  proportional  between  the  focal  radii 
drawn  to  the  same  point. 

107 


108 


LOCI   OF   SECOND   ORDER 


[Chap.  VII. 


Fig.  67 


91.  Directrices.     Let  e  represent   the  eccentricity  and   a  the 
semitransverse  axis  of  ■  an  ellipse  or  an   hyperbola.     The   lines 

drawn  perpendicu- 
lar to  the  transverse 
axis,  one  at  the  dis- 
tance -  to  the  right, 
e 

ct 
and  the  other  -  to 
e 

the  left  of  the  cen- 
ter, are  called  the 
directrices.  In  Figs. 
67  and  68,  the  lines 
DE  and  DiE^  are 
the  directrices. 
The  focus  and  directrix  on  the  same  side  of  the  center  are  said 

to  correspond  to  each  other.     Thus,  DE  corresponds  to  F  and 

D^E,  to  F,. 

92.  A  fundamental  theorem.  If  the  length  of  the  focal  radius  to 
any  point  on  an  ellipse  or  an  hyperbola  is  divided  by  the  distance 
of  the  point  from  the  corre- 
sponding directrix,  then  the  ratio 
so  formed  is  constant  and  equal 
to  the  eccentricity  of  the  curve. 

We  are  to  prove   (Figs.  67 
and  68)  that 

FP^F\P^^ 

PD     PD^       ' 
where  P  is  any  point  on  either 
curve.     From  Art.  50,  we  have 
FP  =  a  +  ex  and  F^P  =  a  —  ex, 
and  from  Fig.  67, 

Pi)  =.  «  _^.  X  and  PA  =--x. 
e  e 

FP     a  +  ex  ,    F,P 

PD='ir-:  =  ''^'''^  pd: 


Hence, 


+  x 


a 

X 

e 


•  =  e. 


Arts.  91-93]       CONSTRUCTION   OF  AN   ELLIPSE 


109 


.  The  theorem  is  proved   for   the  hyperbola  in  the  same  way, 
making  use  of  the  lengths  of  the  focal  radii  (Art.  52)  and  Fig.  68. 

93.  Construction  of  an  ellipse  or  an  hyperbola.  The  theorem  in 
the  preceding  article  furnishes  a  convenient  method  for  consti'uct- 
ing  an  ellipse  or  an  hyper- 
bola when  the  eccentricity 
and  the  distance  from  a 
focus  to  the  corresponding 
directrix  are  known.  For 
example,  to  construct  the 
hyperbola  whose  eccentric- 
ity is  f  and  the  distance 
from  one  focus  to  the  cor- 
responding directrix  is  2, 
let  F  be  one  focus  and  AB 
the  corresponding  directrix 
(Fig.  69),  so  that  the  dis- 
tance QF  is  2.  DraAv  a 
parallel  through  F  to  AB, 
and  lay  off  the  equal  dis- 
tances i^J/and  F2I'  so  that* 


FM^  FM'  ^  3 

QF      QF      2' 

Draw  QM  and  QM'  and  a 

series   of  parallels  to  AB. 

and  QM'  in  L  and  L\  respectively,  and  QF  in  D 

QFM  and  QDL  are  similar,  and  therefore 


Let  one  of  these  parallels  meet  QM 
The  triangles 


FM :  QF::  DL:  QD, 

or  the  ratio  DL  :  QD  is  equal  to  the  given  eccentricity  -|.  With 
F  as  center  and  DL  as  radius,  draw  arcs  of  a  circle  cutting  LL' 
in  H  and  R'.  Then  R  and  R'  are  points  on  the  curve,  since 
FR  :  QD  =  DL  :  QD  =  3:2.  In  this  way  as  many  points  of  the 
curve  can  be  located  as  may  be  desired. 


*  Here,  and  for  the  most  part  throughout  this  chapter,  we  are  not  concerned 
with  directed  segments  or  with  directed  angles. 


110  LOCI  OF  SECOND   ORDER  [Chap.  VII. 

An  ellipse  can  be  constructed  in  a  similar  way. 

By  construction,  the  angle  FQM=  arc  tan  e  and  is,  therefore, 
greater  than  45°  for  the  hyperbola  and  less  than  45°  for  the  ellipse 
(Arts.  50  and  52). 

94.  Two  common  properties.  By  definition  (Art.  53),  the  length 
of  the  focal  radius  to  any  point  on  a  parabola  divided  by  the  dis- 
tance from  the  point  to  the  directrix  is  equal  to  unity.  Compar- 
ing this  statement  of  the  definition  of  the  parabola  with  the 
theorem  in  Art.  92,  we  are  led  to  define  the  eccentricity  of  the 
parabola  as  unity.  Consequently,  the  ellipse,  hyperbola,  and  parab- 
ola have  the  following  important  property : 

(^1)  If  the  length  of  the  focal  radius  to  any  point  on  one  of  these 
curves  is  divided  bi/  the  distance  from  the  point  to  the  corresponding 
directrix,  then  the  ratio  so  formed  is  constant  and  equal  to  the  eccen- 
tricity of  the  cxirve;  this  ratio  is  less  than  1  for  the  ellipse,  equal  to 
1  for  the  parabola,  and  greater  than  1  for  the  hyperbola. 

The  three  curves  have  another  important  property  in  common ; 
namely, 

(B)   They  are  sections  of  a  right  circular  cone  (Part  II,  Art.  159). 

The  circle  is  also  a  section  of  a  right  circular  cone.  The  four 
curves  are  therefore  called  conic  sections,  or  more  briefly,  conies. 

Note.  The  conies  were  originally  studied  by  the  Greeks,  who  used 
property  (J5)  as  a  definition.  Property  {A)  was  probably  known  to  Euclid 
and  his  contemporaries  (300  B.C.),  but  the  earliest  mention  of  it  now  known 
to  exist  occurs  in  the  "  Collections  of  Pappus  "  (100  a.d.). 

EXERCISES 

1.  In  an  ellipse,  given  a  =  3  and  6=2.  Find  c  and  e,  locate  accurately 
the  foci  and  directrices,  find  the  distance  from  a  focus  to  the  corresponding 
directrix ;  write  the  standard  equation  in  rectangular  coordinates,  the  polar 
equation  with  the  pole  at  the  left-hand  focus,  and  the  parametric  equations. 

2.  Find  a  and  b  for  the  hyperbola  constructed  in  Art.  93.  Write  the 
standard  equation  in  rectangular  coordinates,  the  polar  equation  with  the 
pole  at  the  left-hand  focus,  and  the  parametric  equations. 

3.  Make  an  accurate  construction  of  the  ellipse  for  which  e  =  \  and  a  =  6, 
Locate  the  foci  and  the  directrices  and  write  the  three  standard  forms  of 
its  equation  as  in  the  preceding  exercises. 


Arts.  94,  95] 


EQUATION  OF  A  TANGENT 


111 


4.  Construct  the  hyperbola  for  which  a  =  4  and  5  =  5.  Locate  the  foci 
and  directrices  and  write  the  tliree  standard  forms  of  its  equation. 

5.  The  length  of  the  focal  radius  drawn  to  one  extremity  of  the  minor 
axis  of  an  ellipse  is  5  and  the  eccentricity  is  f.  Construct  the  curve 
and  locate  the  foci  and  directrices. 


TANGENTS 

95.  Equation  of  a  tangent  in  terms  of  the  slope.  If  a  straight 
line  meets  a  curve  in  the  points  P  and  Q,  these  points  will  move 
along  the  curve  Avhen  the  line  is  either  rotated  about  some  point 
in  the  plane  or  moved  parallel  to  itself.  If  it  is  possible  to  pass 
continuously  along  the  curve  from  P  to  Q,  it  will  be  possible  to 
move  the  line  in  either  of  the  ways  mentioned  so  as  to  cause  P 
and  Q  to  coincide  in  a  point  R.  The  line  PQ  is  then  tangent  to 
the  curve  at  R,  and  R  is  the  point  of  contact. 

To  find  the  coordinates  of  the  points  of  intersection  of  a  curve 
with  a  straight  line,  it  is  necessary  to  solve  the  equation  of  the 
curve  and  the  equation  of  the  line  simultaneously. 

T7ie  circle.     Let  the  given  curve  be  the  circle 


X-  +  y  =  a^, 

and  take  the  equation  of  the  line  in  the  slope  form 

y  =  mx  +  k. 

Eliminating  y  between  (1)  and  (2), 
we  see  that  the  .^'-coordinates  of  the 
points  of  intersection  are  the  roots 
of  the  equation 

(1  +  »n-2)x2  +  2mkx  +(7^'^ 


The  line  will  move  parallel  to 
itself  when  7c  is  allowed  to  vary,  and 
the  points  of  intersection,  P  and  Q 
(Fig.  70),  will  coincide  when,  and 
only  when,  the  roots  of  equation  (A)  are  equal 


(1) 


Fig.  70 


that  is,  when 


4  (1  +  vi")  {k~  -  a2)  =  4  m^'' 


112  LOCI   OF  SECOND   ORDER  [Chap.  VII. 

Solving  this  equation  for  k,  we  have 


A;  =  ±  a  Vl  +  m^ 

Hence,  when  Ti  has  either  of  these  vahies,  the  roots  of  (A)  are 
equal  and  the  points  of  intersection  of  the  circle  with  the  line 
coincide.     Therefore,  the  lines 

y  =  tnx  ±a'^l  +  ni^  (3) 

are  tangents  to  the  circle  x^  -{-if  ^=  dr. 

The  ellipse.     Similarly,  solving  the  equation  of  the  ellipse 

-,  +  f,  =  ^  (4) 

and  equation  (2)  simultaneously,  we  find  that  the  x-coordinates 
of  the  points  of  intersection  -are  the  roots  of  the  equation 

(a2m2  +  &2)ic2  +  2  ahnkx  +  a^(k'2  -  6^)  =  o.  (B) 

The  roots  will  be  equal,  and  consequently  the  line  a  tangent  to 
the  ellipse,  when 


4  a2  (a^m^  +  ¥)  (k^  -b')  =  4  aSn''k\  or  ^^  =  ±  ^dm^  +  b\ 

Therefore,  the  lines  

y  =  mx  ±  ^ahn^  +  b'^  (5) 

are  tangents  to  the  ellipse  ivhose  equation  is  given  in  (4). 

The  hyperbola.     The  a^coordinates  of  the  points  of  intersection 

of  the  hyperbola  ^       2 

^-•1  =  1  (6) 

a^      If 

with  the  line  y  =  mx  -\-  k  are  the  roots  of  the  equation 

(a2^2  -  62)ic2  ^  2  a^mkx  +  a^{1c^  +  b)^  ^d,  (C) 

and  these  roots  are  equal  when  k  has  either  of  the  values 


^-  =  i:  -Va-m"^  —  ¥. 

Therefore,  the  lines  ^ 

y  =  7nx  ±  V  a2^,2  _  52^  {1) 

are  tangents  to  the  hyperbola  whose  equation  is  given  in  (6). 

The  parabola.     The  ^.'-coordinates  of  the  points  of  intersection 
of  the  parabola  y^  =  42yx  (8) 


Art.  95]  EQUATION   OF  A  TANGENT  US 

with  the  line  y  =  mx  +  k  are  the  roots  of  the  equation 

m-jr-2  +  (2  mk  -  4i?)ic  +  A-^  =  0,  (D) 

and  these  roots  are  equal  wiien  k  has  the  value  — 
Therefore,  the  line  ^.  _  ^^       ,   i> 


m 
m 


(9) 

is  a  tangent  to  the  parabola  y^  =  ijxv. 

As  an  example  of  the  use  of  the  foregoiug  formulas,  we  shall  find  the  equa- 
tions of  the  tangents  to  the  hyperbola 

x2  _  4  2/2  =  36 

which  have  the  slope  f .     Here  a  =  6,  6  =  3,  and  m  =  |.     Substituting  in 
(7),  we  find  the  required  equations  are 

6  y  =  5  X  ±  2i. 
Again,  to  find  the  equations  of  the  tangents  to  the  ellipse 
4  x2  +  9  2/2  :=  36, 

which  pass  through  the  point  (2,  3),  use  equation  fo).     Here  a  =  3,  6  =  2, 
and  we  are  to  find  m  so  that  the  tangents  pass  through  the  given  point. 

Hence  we  must  have  

3  =  2m±  V9m2+4, 

from  which  we  find  

-  6  ±  V61 

m  =  — 

5 

The  required  equations  are  therefore 

,/_3  =  ^l±^(.r-2). 
5 

The  formulas  (A),  (B),  (C),  and  (D)  are  of  frequent  use  in 
what  follows. 

Note  that  properties  of  the  hyperbola,  expressed  by  equations 
involving  the  semiaxes,  can  be  derived  from  the  corresponding 
properties  of  the  ellipse  by  changing  the  sign  of  b^.  Thus,  equa- 
tions (B)  and  (C)  differ  only  in  the  sign  of  b^. 

EXERCISES 

1.    Find  the  equations  of  the  tangents  to  the  following  conies  : 
(«)  2/2  —  4x,  slope  =  J.  (d)    x^  —  4?/ 2  =  36,  passing  through 

(&)  x^  +  y'^  —  16,  slope  =  —  f  -  the  point  (3,  4). 

(c)  9  a:2  +  16  y'^  =  144,  slope  —  —  \.       (e)    a;2  +  4y2  =  36,  perpendicular  to 

6x  —  4?/  +  9  =  0. 


114  LOCI   OF   SECOND   ORDER  [Chap.  VII. 

2.  Eind  the  equations  of  the  common  tangents  to  the  follomng  pairs  of 
conies.     Construct  the  figures. 

(a)  2/2  =  5  X  and  9  x^  +  9  y^  =  16. 

(6)  9  x2  +  16  ?/2  =  144  and  7  x'^  -32y^  =  224. 

(c)  x2  +  2/2  =  49  and  13  x^  +  50  2/2  =  650. 

Suggestion.  Find  the  equations  of  tangents  to  each  conic  in  terms  of  the 
slope  and  then  determine  the  slope  so  that  the  two  equations  shall  be  depend- 
ent (Art.  83). 

3.  Prove  that  two  tangents  to  the  parabola  y^  =  4px  which  are  perpendic- 
ular to  each  other  intersect  on  the  directrix. 

Suggestion.  The  slopes  of  the  tangents  are  negative  reciprocals  of  each 
other.     Hence  their  equations  are 

y  =  mx  +  £-  and  y  = pm. 

in  m 

But  these  lines  intersect  in  a  point  whose  abscissa  is  — p,  whatever  the  value 
of  in.     Construct  a  figure  illustrating  this  exercise. 

4.  Prove  that  two  tangents  to  the  ellipse 1-  ^  =  1  which  are  perpendic- 

ular  to  each  other  intersect  upon  the  circle  x"^  -\-  y"^  =  a^  -{-  b^. 
Suggestion.     The  equations  of  the  two  tangents  are 


y  =zmx+  Vahn^  -\- b^  and  y  =  -  ~  +  /:^/«i±E™!\  _ 

in       \         in  / 

The  point  of  intersection  must  satisfy  both  these  equations ;  hence  the  equa- 
tion of  its  locus  is  found  by  eliminating  m  from  these  equations.     To  do  this, 
remove  the  radicals  from  each  equation  by  transposition  and  squaring.    Thus, 
(y  —  inx)'^  =  a^m'^  +  &2  and  (my  -|-x)2  =  ^2  _|_  52,j^2_ 

Add  these  equations,  member  to  member,  and  divide  by  the  common  factor 
1  +  in^.  The  circle  x-  +  y^  =  a'^  -i-  62  ig  called  the  director  circle  for  the 
ellipse.     Construct  a  figure  illustrating  this  exercise. 

5.  The  locus  of  the  intersection  of  a  pair  of  perpendicular  tangents  to  the 
hyperbola  is  called  the  director  circle  for  the  hjrperbola.  Find  its  equation 
and  show  that  it  is  a  real  circle  only  when  a  >  & ;  it  reduces  to  a  point  for 
the  equilateral  hyperbola,  a  =  b  ;  and  is  imaginary  for  a<ib.  Construct  a 
figure  illustrating  this  exercise. 

6.  If  a  perpendicular  is  dropped  from  either  focus  of  an  ellipse  (or  an 
hyperbola)  upon  a  tangent,  show  that  the  locus  of  its  intersection  with  the 
tangent  is  a  circle  whose  center  coincides  with  the  center  of  the  curve  and 
whose  diameter  is  the  transverse  axis  of  the  curve. 

Suggestion.  For  the  ellipse,  the  equations  of  the  tangent  and  the  perpen- 
dicular through  the  left-hand  focus  are,  respectively. 


Art.  96]      COORDINATES   OF   POINT   OF   CONTACT  115 


[ X  -X-  c\ 

=  mx  +  \/d-m'-  +  h-  and  y  =  —  \ • 

V    m    I 


Hence,  {y  —  mxY  —  ahii^  +  6"^, 

and  {my  +  .r)-  =  c-  =  «2  _  ^2. 

Adding  and  dividing  by  tlie  common  factor  1  +  ??i'-,  we  have 

J.2   _|_   y'l  —    (i2^ 

Tliis  circle  is  called  the  major  auxiliary  circle.  The  equation  of  the  minor 
auxiliary  circle  is  x'^  +  y~  =  &-  (cf.  Art.  61).  Construct  a  figure  illustrat- 
ing this  exercise. 

7.  Show  that  the  locus  of  the  intersection  of  a  tangent  to  y'^  =  ipx  with 
the  perpendicular  from  the  focus  is  the  3"-axis. 

8.  Show  that  the  product  of  the  perpendiculars  from  the  foci  upon  any 
tangent  to  an  ellipse  is  constant  and  equal  to  b'^.  State  and  prove  the  corre- 
sponding property  for  the  hyperbola. 

96.  Coordinates  of  the  point  of  contact.  Equations  (A),  (B), 
(0),  and  (D)  of  the  preceding  article  serve  to  find  the  coordinates 
of  the  point  of  contact  on  a  tangent  having  the  given  slope  m. 
Thus,  for  the  ellipse, 

(ahn''  +  62).x2  +  2  ahnkx  +  a^k""  -b^)=0  and  k^  =  ahn^-\-  h\ 

Therefore,  k\y?  +  2  ahnkx  +  a^m''  =  0. 

The  left  member  of  this  equation  is  a  perfect  square,  as  it  should 
be,  and  gives  for  the  a>coordinate  of  the  point  of  contact 

x= . 

A," 

Since  the  point  of  contact  is  on  the  line  y  =  mx  +  k,  we  have 
y  =  mx-^k=—^  +  k= =  -. 

Therefore,  the  poUds  of  contact  on  the  tangents  having  the  slope  m 

ahn    b'^^ 
are 


T,  j,  ivhere  k  has  the  values  ±  ^ahn^  +  &^. 

Similarly,  for  the  parabola,  making  use  of  equation  (D)  and 
the  corresponding  value  of  k,  the  .c-coordinate  of  the  point  of  con- 
tact is  given  by  the  equation 

m'^x^  —  2px  +  —  =  0, 


116  LOCI   OF   SECOND   ORDER  [Chap.  VII. 

from  which  x=—.     Substituting  in   y  =  mx  -{-!<:,   we    find  that 
y  —       .     Therefore,  the  point  of  contact  on  the  tangent  having  the 

slope  m  is  [—,  — 
\m?    m 

EXERCISES 

1.  Show  that  the  coordinates  of  the  points  of  contact  of  the  tangents  to  the 

hyperbola =  1,  having  the  slope  m,  are( , ),  where  k  has 

a?-     y'^  \     k  k    I 

the  values  ±  \'a'^n^  —  b'^. 

2.  Show  that  the  coordinates  of  the  points  of  contact  of  the  tangents  to 
the  circle  x^  +  y^  =  a^,  having  the  slope  m,  are  |  ~  ^  "^,  —  ],  where  k  has 
the  values  ±aVin^  +  1. 

3.  Find  the  coordinates  of  the  points  of  contact  of  the  tangents  to  the 
conies  in  exercise  1,  Art.  95. 

4.  rind  the  coordinates  of  the  points  of  contact  of  the  tangents  to  the 
pairs  of  conies  in  exercise  2,  Art.  95. 

97.  Equation  of  a  tangent  in  terms  of  the  coordinates  of  the  point 
of  contact. 

First  method.     Let  P(x^,  i/i)  be  the  point  of  contact  of  a  tangent 

to  the  ellipse 

^  +  ^  =  1. 
a^      b^ 

Then,  by  the  preceding  article, 

—  a'^m       J  b^ 

X.  = and  v.  =  — . 

k  ^        k 

Eliminating  k  by  division,  we  have 

Vi  _  —  b"^    .    _    y^x^ 

Xi       a^m '  a'^yi 

Since   the    tangent   passes    through   the    point   of    contact,    its 
equation  is 

2/  -  2/1  =  m(^  -  ^i)  =  ~  ^  (•^'  ~  ^^y  (^) 


Art.  97]     COORDINATES  OF  POINT   OF   CONTACT 


117 


Clearing  of   fractions  and  remembering  that   IP-x-^  -\-  a?yi  =  a^b^, 
since  the  point  of  contact  is  on  the  ellipse,  equation  (1)  reduces  to 


^^1  I  UUl 


1. 


(2) 


For  the  parabola,  if  P{Xi,  y{)  is  the  point  of  contact  of  a  tangent, 

we  have  seen  that  ?/i  =  -^.     Therefore  the  slope  of  the  tangent 
m 
2p 

2p 


is  -~^,  and  its  equation  is 

y  -yi  =  m{x  —  X,) 


(3) 


Qcxi^h,  2/^+t) 


Fig.  71 


Clearing  of  fractions  and  remember- 
ing that  yi^  =  4  j^x^,  we  have 

2/!/i  =  2i>(a?  +  a?i).  (4) 

Second  method.  Let  a  secant  meet 
a  curve  in  the  points  P(x^,  ?/i)  and 
Q(%  +  h,  ?/i  +  k)  (Fig.  71),  so  that 
the  projections  of  the  segment  PQ 
upon  the  X-  and  F-axes  are  respectively  h  and  k.     The  slope  of 

PQ  is  then  -.      The  coordinates  of  P  and  Q  satisfy  the  equation 

of  the  curve.     Hence,  if  the  curve  is  an  ellipse  as  in  the  figure  we 
have 

^  +  lll  =  l    and    (^i+M  +  (li±J^  =  i,  (5) 

Subtracting  the  first   equation  from  the  second,  member   from 
member,  we  obtain  the  equation 


from  vs^hich  we  get 


2  hx,  +  h'  ,   2  ky,  +  k^  ^  ^ 
a^  b^ 

k^  b\2x,  +  h) 

h  a\2y,  +  k) 


(6) 


As  the  secant  rotates  about  P  (cf .  Art.  95),  the  point  Q  approaches 
P  along  the  curve  and  in  the  limit  coincides  with  it,  and  then  the 
secant  becomes  the  tangent  at  P.  But  the  slope  of  the  secant  is 
constantly  equal  to  the  right-hand  member  of  (6).     When  Q  coin- 


118  LOCI  OF   SECOND   ORDER  [Chap.  VII. 

cides  with  P,  both  h  aud  k  are  zero,  and  the  right-hand  member 
of  (6)  gives  the  slope  of  the  tangent  at  P;  that  is, 

m  = -. 

The  equation  of  the  tangent  is  then  found  as  in  tlie  first  method. 

EXERCISES 

1.  Write  the  equation  of  the  tangent  to  the  ellipse  3  x'-^  +  4y"  =  19  at 
the  point  (1,  2). 

2.  Show  that  the  equation  of  the  tatfgent  to  the  hyperbola  - — ^—1 

Or'      b- 

at  the  point  (xi,  ?/i)  is  ?^_Mi=  i. 

3.  Write  the  equation  of  the  tangent  to  the  hyperbola  2  x^  —  ?/2  —  14  at 
the  point  (.3,  —  2). 

4.  Find  the  equations  of  the  tangents  to  the  ellipse  16  x^  +  25  y-  =  400 
which  pass  through  the  point  (3,  4). 

5.  Write  the  equation  of  the  tangent  to  the  parabola  y^  =  6  x  at  the 
point  (6,  -  6). 

6.  Find   the   angle    which   the   ellipse    4  x-  +  2/'-  =  5    makes   with   the 
parabola  y'^  —  8  x  at  ?i  point  of  intersection. 

Suggestion.  Find  the  equation  of  the  tangent  to  each  curve  at  a  point 
of  intersection  and  then  find  the  angle  which  one  tangent  makes  with  the 
other. 

7.  Show  that  the  equation  of  the  tangent  to  the  circle  x^  -fy^  =  a-  at  the 
point  (xi,  yi)  is  xxi  +  yyi  =  a^. 

8.  Show  that  the  length  of  a  tangent  to  the  circle  x-  +  y^  =  a-,  included 
between  the  point  of  contact  and  the  point  (xo,  2/2),  is  Vx-r^  f-  yz^  —  a'^. 

9.  Prove  that  the  circles  whose  equations  are  x'^  +  y^  —  8  x  +  4  ?/  +  7  =  0 
and  x^  +  2/2  —  10  X  —  6  2/  +  21  =  0  intersect  at  right  angles. 

Suggestion.  Show  that  the  square  of  the  distance  between  the  centers  is 
equal  to  the  sura  of  the  squares  of  the  radii. 

10.  Using  the  second  method  of  Art.  97,  find  the  equations  of  the  tan- 
gents to  the  following  curves  at  the  points  designated  : 

(a)  2/2  =  x3,  at  (4,  8).  (6)  y  =  x2(x- 1),  at  (2,  4). 

(c)  ys  =  x%  at  (8,  4).  (d)  y^=x(x  -l){x  -  2),  at  (8,  V6). 

Draw  each  curve, 


Arts.  98,  99]    TANGENT   LENGTH,  NORMAL  LENGTH      119 


98.  Normals.  Given  any  curve,  the  line  drawn  perpendicular 
to  a  tangent  at  the  point  of  contact  is  called  the  normal  to  the 
curve  at  the  point  of  contact. 

Let  P(.x"i,  2/i)  ^6  the  point  of  contact  and  m  the  slope  of  the  tan- 
gent at  P,  then  the  equation  of  the  normal  is 


y  -yi  = {x-x,). 


(1) 


For  example,  the  slope  of  the  tangent  to  the  ellipse 

^-4-^  =  1 

at  the  point  Cx^,  Wj)  is —  (Art.  97).     Hence  the  equation  of 

the  normal  at  (.v,,  y^)  is 


(2) 


99.  Tangent  length,  normal  length,  subtangent,  subnormal.  Con- 
nected Avith  every  point  on  a  curve  there  is  a  special  triangle 
whose  sides  are  respec- 
tively the  tangent  at 
the  point,  the  normal 
at  the  point,  and  the 
X-axis.  In  Fig.  72, 
PTN'\^  such  a  triangle, 
where  FT  is  the  tan- 
gent at  P,  PN  is  the 
normal  at  P,  and  TN 
is  the  X-axis.     PT  is 

called  the  tangent  length  and  PN  the  normal  length.  DP  is  the 
ordinate  of  P,  PT  is  the  projection  of  PT  on  the  X-axis  and  is 
called  the  subtangent,  1)1^  is  the  projection  of  PiVon  the  X-axis 
and  is  called  the  subnormal. 

Let  the  coordinates  of  P  be  ccj,  y^  and  the  inclination  of  the 

tangent  be  ^,  then  DP  —  y^  and  the  angle  DPN  —  angle  DTP. 

Therefore,  dot      i  ,  i        • 

PT  =  I  yi  cosec  ^  \, 

PX=  |?/i  sec  ^1, 


Fig.  72 


120  LOCI   OF   SECOND   ORDER  [Chap.  VII. 

Z>r=|2/iC0tc/>I, 

where  the  bars  indicate  positive,  or  absolute,  value. 

EXERCISES 

1.  Write  the  equation  of  the  normal  to  the  circle  at  the  point  (a:i,  yi). 
Note  that  the  normal  passes  through  the  center  of  the  circle. 

2.  Write  the  eqiiation  of  the  normal  to  the  parabola  at  point  (xi,  yi). 
The  equation  of  the  normal  to  the  hyperbola  at  (xi,  yi). 

3.  The  point  (3,  2)  lies  on  the  ellipse  ofi  +  4?/2  =  25.  Find  the  tangent 
length,  the  normal  length,  the  subtangent,  and  the  subnormal,  at  this  point. 
Construct  the  figure. 

4.  Find  the  equation  of  the  normal  to  the  parabola  ?/-  =  8  x  which  is 
parallel  to  the  line  2a;  +  3?/  =  10. 

5.  Prove  that  the  normal  to  the  ellipse  or  the  hyperbola  at  the  point 
(xi,  2/1)  meets  the  X-axis  at  a  distance  e-xi  from  the  center. 

6.  Show  that  the  subnormal  to  the  parabola  y'^  —  4:px  is  constant  and 
equal  to  2  p. 

7.  The  line  3  x  +  8  j/  =  25  is  tangent  to  the  ellipse  x'^  +  4  ?/2  =25.  Find 
the  coordinates  of  the  point  of  contact  and  write  the  equation  of  the  normal 
at  this  point. 

8.  The  line  mx  —  iy  =  1  is  tangent  to  the  hyperbola ^  =  1.     Find 

9        4 

m  and  compute  the  subtangent  and  subnormal  for  the  point  of  contact. 

100.  Reflection  properties.  The  three  theorems  that  follow 
express  what  are  known  as  reflection  properties. 

Theorem  I.  The  angle  formed  by  the  focal  radii  drawn  to  any 
point  of  an  ellipse  is  bisected  by  the  normal  at  that  point. 

The  equation  of  the  normal  at  the  point  (x^,  y{)  is  given  in  (2), 
Art.  98.  From  this  equation  we  find  the  intercept  on  the  X-axis 
is  (Fig.  73)  (a^-b^)x,      c^x,      a^e^x,        „ 

a^  a^  a- 

But  FO  =  OFi  =  ae,  and  therefore 

FN  =  ae  +  e^x^     and     NF^^  =  ae  —  e\. 
The  ratio  of  FN  to  NF^  is,  therefore, 

FN _a-\-ex^_  FP 


NF^      a  -  ex^      F,P 


(Art.  50). 


Art.   100] 


REFLECTION  PROPERTIES 


121 


Hence,  we  have  sliowu  that  the  normal  at  P  divides  the  base  of 
the  triangle  PFF^  into  two  segments  which  are  proportional  to 
the  adjacent  sides.     Therefore  the  normal  bisects  the  angle  FPF^ 


Fig.  73 

A  ray  of  light  from  either  focus  of  an  ellipse  is  reflected  from 
the  curve  to  the  other  focus. 

Theorem  II.  The  angle  formed  by  the  focal  radii  drawn  to  any 
point  of  an  hyperbola  is  bisected  by  the  tangent  at  that  point. 

The  equation  of  the  tangent  at  {x-^,  y^)  is 


a"       b- 


Hence  the  intercept  on  the  X- 
axis  is  (Fig.  74), 

0T=-- 


We  can  now  show  that 

FT  ^  PF 
TF,  ~  PF, ' 

For, 

and  (Art.  52) 


Fig.  74 


FT  =  ae  +  - 

X, 


TF,  =  ae- 


PF  =  ex- 1  +  a     ,     PFi  =  ex^ 


122 


LOCI   OF   SECOND   ORDER 


[Chap.  VII. 


Hence  the  tangent  at  P  divides  the  base  of  the  triangle  PFF^^  into 
two  segments  which  are  proportional  to  the  adjacent  sides,  and 
therefore  bisects  the  angle  FPF^. 

Theorem  III.  Any  tangent  to  the  parabola  if  =  4: iix  bisects  tJie 
angle  formed  by  the  focal  radius  drawn  to  the  point  of  contact  and  a 
line  through  the  p)oint  of  contact  jKtrallel  to  the  X-axis. 

The  equation  of  the  tangent 
to  the  parabola  at  the  point 
P(x„  y,)  (Fig.  75)  is  (Art.  97) 

Hence,  the  tangent  meets 
the  X-axis  at  the  point 
T  =  (  -  x„  0).  Therefore  TO 
=  OD,  and,  if  EG  is  the 
directrix  and  F  the  focus, 
E0=  OF.  Hence,  TF  = 
ED.  But  ED  =  FP  (defi- 
nition of  the  parabola),  and, 
consequently,  TFP  is  an  isosceles  triangle.  The  angles  PTF 
and  FPT  are  therefore  equal.  If  PG  is  parallel  to  the  X-axis, 
the  angles  PTF  and  TPG  are  equal.  Consequently,  the  tangent 
PT  bisects  the  ana:le  FPG. 


Fig.  75 


EXERCISES 

1.  Two  parabolas  have  a  common  axis  and  a  common  focus,  and  extend 
in  opposite  directions.     Show  that  tliey  intersect  at  right  angles. 

2.  Given  the  focus  and  the  vertex  of  a  parabola,  but  not  tlie  constructed 
curve,  show  how  to  draw  the  two  tangents  through  a  given  point  P. 

Suggestion.  Let  F  be  the  focus  and  A  the  vertex.  Draw  the  line  AF 
and  construct  the  directrix.  With  P  as  center  and  PF  as  radius,  draw  a 
circle  meeting  the  directrix  in  D  and  D\.  The  perpendicular  bisectors  of  DF 
and  D\F  are  the  required  tangents.  The  tangents  thus  constructed  meet  the 
perpendiculars  to  the  directrix  at  D  and  Z>i  in  the  points  of  contact. 

3.  Show  tliat  an  ellipse  and  an  hyperbola  having  the  same  foci  intersect 
at  riglit  angles. 

4.  Having  given  the  length  of  the  transverse  axis  and  the  distance  between 
the  foci  of  an  ellipse,  or  an  hyperbola,  show  how  to  construct  the  tangents  to 
the  curve  from  a  given  point  P. 


Arts.  101,  102]         CONJUGATE   DIAMETERS 


123 


Suggestion.  Let  AB  be  the  given  transverse  axis.  Locate  tlie  foci  on 
AB  at  the  points  F  and  Fi.  With  P  as  center  draw  a  circle  through  the 
nearer  focus  Fi,  and  witla  F  as  center  and  AB  as  radius,  a  second  circle 
meeting  the  first  in  the  points  D  and  Di.  The  perpendicular  bisectors  of 
DFi  and  DiFi  are  the  required  tangents.  These  tangents  meet  the  lines  FD 
and  FDi  in  the  points  of  contact. 

5.  Why  is  light  emanating  from  the  focus  of  a  parabolic  mirror  reflected 
in  parallel  rays  ?     What  use  is  made  of  this  fact  ? 


^(•ri,y,) 


DIAMETERS 

101.   Definition.     Any  line  through  the  center  of  a  circle,  an 
ellipse,  or  an  hyperbola  is  called  a  diameter  of  the  curve. 

Any  line  perpendicular 
to  the  directrix  of  a  parab- 
ola is  called  a  diameter 
of  the  curve.  That  di- 
ameter of  the  parabola 
which  passes  through  the 
focus  is  the  axis. 

The  circle,  ellipse,  and 
hyperbola  are  called  cen- 
tral conies ;  the  parabola 
has  no  center,  and  is  there- 
fore called  noncentral. 


Fig.  76 


102.   Conjugate  diameters.      If   P  (.^\,  ,?/,)    is   any  point  on  a 
central  conic  and  PT  is  the  tangent  at  P,    then  the  diameter 
through   P  and   the    diameter    parallel    to    PT   are   called  con- 
jugate   diameters.      Thus,     PO    and 
QO  are  conjugate   diameters  (Figs. 
76  and  77). 

Theorem.     If  m   and  m'    are  the 
slopes  of  a  pair  of  conjugate  diameters. 


then 


mm'  =  T-^, 


Fig.  77 


according  as  the  conic  is  an  ellipse  or 
an  hypierhola. 


124  LOCI   OF   SECOND   ORDER  [Chap.  VII. 

In  either  Fig.  76  or  Fig.  77,  the  slope  of  PO  is 

V\ 

a-i 

^^^        the  slope  of  QO,  =  the  slope  of  PT,  =  m'  =  T^\ 

according  as  the  conic  is  an  ellipse  or  an  hyperbola.    Consequently, 

'       -r  b' 
mm  =  T  —' 

cr 

Since  the  product  of  the  slopes  is  independent  of  the  coordinates 
of  P,  it  follows,  in  case  of  the  ellipse,  that  the  tangent  at  Q  is 
parallel  to  the  diameter  PO. 

EXERCISES 

1.  Given  any  diameter  of  an  ellipse  or  an  hyperbola,  construct  its  con- 
jugate diameter. 

2.  The  point  ( "^-^ ,  1 )  lies  on  the  ellipse  4  x^  +  9  y^  =  36.  Find  the  equa- 
tions of  the  diameter  through  the  point  and  its  conjugate  diameter. 

3.  The  point  (xi,  yi)  lies  on  the  ellipse  —  +  ^  =  1.      Show  that  the  equa- 

a^      52 

tions  of  the  diameter  through  the  point  and  its  conjugate  diameter  are,  re- 
spectively, 

2/ix-xi2/  =  Oand^  +  M  =  o. 

4.  The  point  (xi,  wi)  lies  on  the  hyperbola^ —  ^  =  1.  Find  the  equa- 
tions  of  the  diameter  through  the  point  and  its  conjugate  diameter. 

5.  If  a  diameter  of  the  ellipse  —  -\-^^  =  1  meets  the  curve  in  the  point 
(xi,  2/1),  show  that  the  conjugate  diameter  meets  the  curve  in  the  points 

_Ml,MUndf^,-^V 
b      a   I  \b  a  ) 

6.  Prove  that  the  sum  of  the  squares  of  any  two  conjugate  semidiameters 
of  an  ellipse  is  equal  to  the  sum  of  the  squares  of  the  semiaxes. 

7.  Show  that  conjugate  diameters  of  a  circle  are  always  at  right  angles  to 
each  other. 

8.  What  is  the  relation  between  the  slopes  of  conjugate  diameters  of  the 
equilateral  hyperbola  (5  =  a)  ? 


Art.  103] 


THE  LOCUS   OF  MIDDLE   POINTS 


125 


9.  Two  chords  are  drawn  from  any  point  of  an  ellipse  or  an  hyperbola 
to  the  extremities  of  a  diameter.  Show  that  the  diameters  bisecting  these 
chords  are  conjugate  diameters. 

Suggestion.  Let  the  coordinates  of  the  point  be  Xz  and  yn,  and  the  coor- 
dinates of  one  extremity  of  the  diameter  be  Xi  and  ?/i.  The  coordinates  of 
the  other  extremity  are  then  —  xi  and  —  y-^.     Show  that  the  product  of  the 

slopes  of  the  diameters  bisecting  the  chords  is 
for  the  hyperbola. 


—  ,  for  the  ellipse,  and  -| 

d^  d^ 


10.  Show  that   the   area   of    a    parallelogram    inscribed   in   the    ellipse 

1-^=1  whose  diagonals  are  conjugate  diameters  is  2  ah. 

or      b'^ 

Suggestion.     Let  0  be  the  center  of  the  ellipse,  P(xi,  yi)  and  'Q,  two  adja- 
cent vertices  of  the  parallelogram.     The  coordinates  of  Q  may  then  be  taken 

as  (  — , —]  (exercise  6).     The  ai-ea  of  the  parallelogram  is  four  times 

\  b  a  I 

the  area  of  the  triangle  POQ. 

11.  Prove  that  the  axes  of  an  ellipse  or  an  hyperbola  form  a  pair  of  con- 
jugate diameters. 

Suggestion.     As  the  slope  of  one  diameter  approaches  zero,  what  does 
the  slope  of  the  conjugate  diameter  approach  ? 

12.  Can  a  pair  of  conjugate  diameters  of  an  ellipse  or  an  hyperbola  ever 
be  at  right  angles  unless  they  are  the  axes  ?     Why  ? 


103.    The  locus  of  the  middle  points  of  a  system  of  parallel  chords. 

Theorem.     The  locus  of  the  middle  points  of  a  system  ofjmrallel 
chords  of  any  conic  is  a 
diameter  of  the  conic. 

Let  y  =  mx  +  A."  be 
the  equation  of  any 
line  meeting  the  conic 
in  the  points  Pi  and  P.,- 
If  the  conic  is  an  el- 
lipse, as  in  Fig.  78,  the 
a^coordinates  of  the 
points  Pi  and  P2  are  the 

roots  of  equation  (B),  ^ig  78 

Art.  95,  and  if  x^  and  X2 
represent  these  roots,  then  the  ic-coordinate  of  the  middle  point 


126      .  LOCI   OF   SECOND   ORDER.  [Chap.  VII. 

of  the  chord  P1P2  is  given  by  the  equation 

„j ^1  ~l~  ^2 

But  the  sum  of  the  roots  of  (B)  is Hence, 


and  since  the  middle  point  is  on  the  chord, 
y'  =  mx  -\-  k  = 


ahn"^  +  6^ 


Therefore,  whatever  value  is  given  to  Tc,  the  coordinates  of  the 
middle  point,  x'  and  y' ,  satisfy  the  equation 

y  =  -^~^-  (1) 

But  (1)  is  the  equation  of  a  straight  line  passing  through  the 
center  0  and  is  therefore  a  diameter  of  the  curve.  The  line  PO 
represents  this  diameter. 

If  li  =  0,  the  line  P1P2  assumes  the  position  QS,  which  is  also  a 
diameter  of  the  curve.     Since  the  product  of  the  slopes  of  PO 

and  QS  is  —  — ,  these  diameters  are  conjugate  to  each  other. 
a- 

Combining  this  result  with  the  theorem  in  the  preceding  article, 

we  conclude  that  all  the  chords  parallel  to  PO  are  bisected  by  QS. 

The  theorem  is  proved  for  the  other  conies  in  a  similar  way, 

making  use  of  equations  (A),  (C),  and  (D)  of  Art.  95. 

EXERCISES 

1.  Find  the   equation  of  the  diameter  of  the  hyperbola  x^  —  8  i/^  =  96 
bisecting  all  the  chords  parallel  to  the  line  3  a;  —  8  2/  =  10. 

2.  Find  the  equation  of  the  diameter  of  the  parabola  y-  =.Qx  bisecting  all 
the  chords  parallel  to  the  line  x  +  3  ;/  =  8. 

3.  What  is  the  equation  of  the  chord  of  the   ellipse  9  x^  +  36  ?/2  =  324 
which  is  bisected  by  the  point  (4,  2)  ? 

4.  Find  the  equation  of  the  chord  of  the  ellipse  13  x^  +  11  ifi  =  113  which 
passes  through  the  point  (1,  3)  and  is  bisected  by  the  diameter  2  y  =  3  x. 


Art.  104]  DEFINITIONS  127 

5.  What  is  the  equation  of  the  chord  of  the  parabola  y^  =  6  x  which  is 
bisected  by  the  point  (4,  3)  ? 

6.  Find  the  equation  of  the  diameter  of  the  hyperbola  —  _  ^  =  1  which 
bisects  all  the  chords  of  slope  m.  ^ 

7.  Prove  that  the  diagonals  of  any  circumscribing  parallelogram  to  an 
ellipse  form  a  pair  of  conjugate  diameters  of  the  curve. 

ScGGESTiox.     Let  m  and  n  represent  the  slopes  of  a  pair  of  adjacent  sides 
of  the  parallelogram.     Show  that  the  slopes  of  the  diagonals  are 

mki  —  nk        ■,    mk]  +  nk 
k\  —  k  A'l  +  k 

where  k'^  =  d-m-  +  i>-   and   k{^  =  aP-n-  +  h-.     The   product   of   the  slopes  is 
therefore .     The  diagonals  pass  through  the  center  of  the  ellipse  and 

are  therefore  diameters. 

8.  Prove  exercise  9  of  the  preceding  article  by  showing  that  the  diameter 
bisecting  one  chord  is  parallel  to  the  other. 


POLES   AND   POLAR   LINES 

104.  Definitions.  It  has  been  shown  that  the  equation  of  a 
tangent  to  a  conic  can  be  expressed  in  terms  of  the  coordinates 
of  the  point  of  contact.  For  example,  we  saw  in  Art.  97  that  the 
equation  of  the  tangent  to  the  parabola  at  the  point  P(x^,  y^)  is 

yi/,  =  22y(x-{x,).  (1) 

This  equation  is  the  equation  of  a  straight  line,  whatever  values 
are  given  to  x^  and  yi,  and  is  the  equation  of  a  tangent  to  the 
parabola  only  when  the  point  P{x^,  yx)  is  on  the  curve. 

In  general  (1)  is  the  equation  of  a  straight  line  called  the 
polar  line  of  P{xi,  yi)  with  respect  to  the  parabola  y'^  =  4,px. 
The  point  Pix^,  y-^  is  called  the  pole.  If  the  pole  is  on  the  curve, 
the  polar  line  is  tangent  to  the  curve  at  the  pole. 

Similarly,  the  equation  of  the  polar  line  of  any  point  with 
respect  to  any  conic  can  be  written  at  once.  For  example,  the 
equation  of  the  polar  line  of  P{X],  y^)  with  respect  to  the 
ellipse  is 

We  are  led,  then,  to  the  following  definition : 


128 


LOCI  OF  SECOND  ORDER 


[Chap.  VII. 


Tlie  polar  line  of  P {x^^,  y^)  with  respect  to  a  given  conic  is  that 
line  ivhose  equation  has  the  same  form  as  the  equation  of  the  tangent 
to  this  conic  when  P  (x^^,  ?/i)  is  thep)oint  of  tangency. 

As  an  example,  we  will  write  the  equation  of  the  polar  line  of  (1,  3)  with 

respect  to  the  circle  x'^  +  y^  =  4.     Here  the  equation  of  the  polar  line  of  any 

point  (xi,  ?/i)  is 

^xi  +  2/2/1  =  4. 

Hence,  the  polar  line  of  (1,  3)  is 

X  4-  3  2/  =  4. 

The  student  should  draw  the  figure  illustrating  this  example. 

As  a  second  example,  find  the 
coordinates  of  the  pole  of  5  x  —  4  ?/ 
+  20  =  0  with  respect  to  the  ellipse 


+ 


yA 


X^ 


25      16 


Here  the  polar  line  of  any  point 
(•^■i,  2/1 )  is 


x:/;i 
25 


16 


Fig.  79 


^  =  5^^  y^  =  - 

25  16 


If  this  equation  and  the  given  equa- 
tion are  the  equations  of  the  same 
straight  line,  we  must  have   (Art. 
83) 
4  k,  and  -  1  =  20  k, 


where  k  is  the  common  ratio.     From  these  equations,  we  find  that  Xi  =—  ^^ 
and  2/1  =  V-     Hence  the  required  pole  is  the  point  (  —  -\5.^  J/)  (Fig.  79). 


EXERCISES 

1.  Write  the  equation  of  the  polar  line  of  each  of  the  following  points  : 

1.  (1,-2)  with  respect  to  x^  +  4  y^  =  16. 

2.  (6,  —  4)  with  respect  to  j/^  =  4  x. 

3.  (—  2,  2)  with  respect  to  5  x^  —  8  y^  =  24. 

4.  (2,  —  3)  with  respect  to  5  x2  +  4  ?/2  =  10. 

2.  Find  the  coordinates  of  the  pole  of  the  line  3  x  —  2  y  =  5  with  respect 
to  the  circle  x'-^  +  2/^  =  25. 

3.  What  are  the  coordinates  of  the  pole  of  5  x  +  4  y  =  7  with  respect  to 
the  ellipse  x^  +  2  ?/2  =  10  ? 

4.  Find  the  coordinates  of  the  pole  of  the  line  x  —  ?/  =  10  with  respect  to 
the  parabola  2/^  =  8  x. 


Akt.  105]       GEOMETRIC   PROPERTIES   OF  POLES 


129 


5.  What  are  the  coordinates  of  the  pole  of  the  line  Ax  +  By  +  C  =  0 
with  respect  to  the  hyperbola  - —  ^  =  1  ? 

6.  Through  the  point  (.ri,  yi)  a  line  is  drawn  parallel  to  the  polar  line  of 

the  point  with  respect  to  the  ellipse  ^  +  ?^  =  1.      What  are  the  coordinates 
of  the  pole  of  this  parallel  ?  a       b^ 

105.    Geometric  properties  of  poles  and  polar  lines.     A  point  is 
outside  a  conic  when  two  tangents  can  be  drawn  from  the  point  to 
the  conic.     A  point  is 
inside  a  conic  when  no 
tangents  can  be  drawn 
from  it  to  the  conic. 

Theorem  I.  If  the 
jjoint  (x-^,  2/i)  is  outside 
a  conic,  its  polar  line 
with  respect  to  the 
conic  passes  through  the 
points  of  contcLct  of  the 
tangents  draivn  from 
the  point. 

Let  the  conic  be  the 
ellipse;  the  proof  is 
similar  for  the  other 

conies.  Let  Bix^,  1/2)  a^^cl  C(xs,  y^)  be  the  points  of  contact  of 
tangents  drawn  from  A(x^,  y^  (Fig-  80)-  The  equations  of  the 
tangents  at  B  and  C  are,  respectively, 


^  +  m  =  l  and^  +  #  =  l. 


a^ 


Since  the  tangents  pass  through  A,  the  coordinates  of  A  satisfy 
each  of  the  above  equations.     Hence, 


^2  I  .M? 

ce         ¥ 


1  and  ^j^3  ^  M?  =  1. 
a"        62 


But  these  same  equations  result  from  substituting  the  coordinates 
of  the  points  B  and  C  in  the  equation  of  the  polar  line  of  A  with 
respect  to  the  ellipse.  Consequently  the  polar  line  of  A  passes 
through  B  and  C. 


130 


LOCI   OF   SECOND   ORDER 


[Chap.  VII. 


Exercise.  Prove  Theorem  I  for  each  of  the  other  three  conies. 

Theorem  II.  If  P  and  Q  are  two  points  in  the  plane  such  that 
the  polar  line  of  P  icith  respect  to  a  given  conic  passes  through  Q,  then 

the  polar  line  of  Q  jJctsses  through  P. 
Take  the  parabola  y^  =  4pa;  (Fig. 
81)  for  the  given  conic.  A  similar 
proof  establishes  the  theorem  for  the 
other  conies.  Let  the  coordinates 
of  P  be  Xi,  i/i  and  the  coordinates 
of  Q,  X2,  2/2-  The  equation  of  the 
polar  line  of  P  is,  then, 


Fig.  81 


Since  this  line  passes  through  Q,  we 

2/22/1  =  2p(a;2  +  a-i). 

But  this  same  equation  results  from  substituting  the  coordinates 
of  P  in  the  equation  of  the  polar  line  of  Q.  Hence,  the  polar 
line  of  Q  passes  through  P. 

Exercise.   Prove  Theorem  II  for  each  of  the  other  three  conies. 
Theorem  III.     If  a  line  through  the  pole  A  meets  the  p>olar  line 
in  C  and  the  conic  in  B  and  D,  then 

AB^     AD 
BC         DC 

Let  the  coordinates  of  A  (Fig.  82)  be  x^,  y^  and  the  coordinates 
of  G  be  x^,  2/2.  The  coordinates  of  the  point  B,  dividing  the  seg- 
ment AC  in  the  ratio  AB  :  BC=  r,  are 


Xi  +  7U*2 


l  +  r 

But  B  lies  on  the  conic, 
the  figure, 


and 


y 


Ih  +  'ilh 
l  +  r 


(Art.  IT). 


Hence,  if  the  conic  is  an  ellipse  as  in 


(x-i  +  rx^y-  ,  (yi  +  ry.y  ^  ^ 
a\l  +  rf       h\l  +  rf 

Expanding  and  arranging  according  to  the  powers  of  r,  we  have 


1     2/2^    _  -J^  V.2  _j_  2!^^  _|_  2/1  ?/2 


iV  + 


+ 


1  Wo. 


Art.  105]      GEOMETRIC   PROPERTIES  OF  POLES 


131 


We  should  have  been  led  to  the  same  equation  had  we  taken 
AB:DC=r.  Hence  the  roots  of  this  equation  are  the  ratios  in 
which  the  curve  points  divide  the  segment  AC.     But,  since  C  is 


Fig.  82 


on  the  polar  line  of  A,  the  coefficient  of  r  vanishes.     Hence  the 
roots  are  equal  but  opposite  in. sign ;  that  is, 

AB^     AD 

BC  ~      DC 

Four  points  A,  B,  C,  and  D  situated  on  a  straight  line  and  such 


constitute  a  harmonic  range.     The  segment  AC 


that^  =  -^i^ 
BC         DC 

is  said  to  be  divided  harmonically  by  B  and  D. 

Exercise.     Prove  Theorem  III  for  the  parabola  and  for  the  circle. 

Theorem  IV.  The  polar  line  of  a  focus  of  a  conic  is  the  corre- 
sponding directrix. 

We  shall  establish  the  theorem  for  the  case  of  an  ellipse,  leav- 
ing the  remaining  cases  as  an  exercise  for  the  student.  The 
coordinates  of  the  right-hand  focus  of  an  ellipse  are  x  =  ae  and 
y  =  0.  Substituting  these  for  x^  and  ?/i  in  the  general  equation 
of  the  polar  line  with  respect  to  the  ellipse,  we  have,  as  the  polar 

line  of  the  focus, 

a 
X  —  -• 

6 


132 


LOCI   OF   SECOND    ORDER 


[Chap.  VII. 


But  this  is  the  equation  of  the  right-hand  directrix. 

Exercise.  Prove  that  the  directrix  of  a  parabola  is  the  polar  line  of  the 
focus  ;  that  either  directrix  of  an  hyperbola  is  the  polar  line  of  the  corre- 
sponding focus. 

Theorem  V.  Tlie  line  joinmg  a  focus  of  a  conic  to  the  intersec- 
tion of  any  tivo  tangents,  bisects  one  of  the  angles  formed  by  the  focal 
radii  draivn  to  the  points  of  contact  of  the  tangents. 

In  Fig.  83,  let  B  and  D  be 
the  points  of  contact  of  tangents 
from  Q,  F  being  a  focus.  Draw 
the  line  BD  and  let  it  meet  the 
directrix  corresponding  to  F  in 
A,  and  the  line  QF  in  C.  Draw 
the  lines  BE  and  DG  perpen- 
dicular to  the  directrix.  Since 
F  is  the  pole  of  the  directrix 
(Theorem  IV)  and  Q  is  the  pole 
of  the  line  BD  (Theorem  I),  it 
follows  that  QF  is  the  polar  line 
of  A  (Theorem  II).  Therefore 
we  have  the  following  equations  : 

(Theorem  III,  CD  =  -  DC),    . 


Fig.  83 


AB 
BC 

EB 
AB 

BF 
EB 


AD 


CD 

GD 
AD 

DF 
GD 


(Similar  triangles), 


(Eccentricity,  property  A,  Art.  94). 


Equating  the  product  of  the  right-hand  members  of  these  equa- 
tions to  the  product  of  their  left-hand  members,  we  obtain 

BF^DF 
BC     CD' 

Hence  the  point  C  divides  the  side  BD  of  the  triangle  BFD  in 
the  ratio  BF :  DF,  and  consequently  the  line  QF  bisects  the 
angle  BFD. 


Art.  106]  ASYMPTOTES   OF  HYPERBOLA  133 


EXERCISES 

1.  Show  how  Theorem  II  can  be  used  to  construct  the  pole  of  any  line 
with  respect  to  a  given  conic. 

SuGGESTiox.  Construct  the  polar  line  of  any  two  points  on  the  given  line. 
"Where  do  these  intersect  ? 

2.  Two  lines  are  drawn  through  a  point  P.  The  poles  of  these  lines  with 
respect  to  any  conic  are  the  points  R  and  Q.  Show  that  BQia  the  polar  line 
of  P  with  respect  to  the  same  conic. 

3.  Given  any  two  lines  in  the  plane  such  that  the  first  passes  through  the 
pole  of  the  second  with  respect  to  any  conic.  Show  that  the  second  passes 
through  the  pole  of  the  first. 

4.  Prove  that  the  intereection  of  any  two  tangents  to  an  ellipse  or  an 
hyperbola  is  equidistant  from  the  four  focal  radii  that  can  be  drawn  to  the 
points  of  contact. 

5.  Show  that  the  intersection  of  any  two  tangents  to  a  parabola  is  equi- 
distant from  the  focal  radii  to  the  points  of  contact  and  the  diameters  through 
the  points  of  contact. 

6.  In  Fig.  75,  Art.  100,  let  PO  meet  the  directrix  in  J/,  and  PT  meet 
the  vertical  tangent  in  Q.     Show  that  QF  '\s  the  polar  line  of  M. 

SYSTEMS    OF   CONICS 

106.  The  asymptotes  of  the  hyperbola.  It  has  already  been 
noticed  that  the  hyperbola  has  asymptotes  (Art.  67).  This  is  a 
characteristic  property  of  the  hyperbola.  No  other  conic  has 
asymptotes. 

Solving  the  standard  form  of  the  equation  of  an  hyperbola  for 
y,  we  have 

y=±-  ■\/x^  —  al 
a 

Hence,  as  x  increases  indefinitely,  y  approaches  nearer  and  nearer 

to  the  values  ±  —  •     Therefore 
a 

!/=±~  (1) 

are  the  equations  of  the  asymptotes. 

The  equations  of  the  asymptotes  can- be  written 

^-^=0,  (2) 


134 


LOCI   OF   SECOND   ORDER 


[Chap.  VII. 


since   the   coordinates    of   any  point   on   either   asymptote   will 
satisfy  equation  (2)  (cf.  Art.  85). 

The  equation  of  a  tangent  to  the  hyperbola  in  terms  of  the 
slope  is  (Art.  95,  Eq.  7) 

y  z=  mx  +  -yjahii^  —  h"^.  (3) 

If,  in  this  equation,  m  is  taken  as  the  slope  of  either  asymptote ; 

namely,   ± -,  the  equation  becomes  y  =  ± —     For  this' reason, 
a  « 

the  asymptotes  are  often  spoken  of  as  tangents  to  the  Jiyperbola,  the 

points  of  contact  being  infinitely  distant. 

Since  an  asymptote  passes  through  the  center,  it  is  a  diameter 

of  the  hyperbola  (Art.  101).     The  product  of  the  slopes  of  a  pair 

of  conjugate   diameters  of  the  hyperbola  is   —   (Art.  102),  and 

therefore  each  asymptote  is  its  own  conjugate  diameter,  or  in  other 
words,  an  asymptote  is  a  self-conjugate  diameter  of  the  hyperbola. 


107.    Conjugate  hyperbolas.     The  two  hyperbolas 
a2 


-t'^1,  and^-r  =  _l 

'    ¥  ce    b- 


are  called  conjugate  hyperbolas.  The  transverse  and  conjugate 
axis  of  the  one  are  respectively  the  conjugate  and  transverse  axis 
of  the  other.     Either  hyperbola  is  conjugate  to  the  other,  but  it 


Fig.  84 


Arts.  107,  108]      CONCENTRIC  HYPERBOLAS  135 

is  convenient  to  speak  of  the  first  as  the  primary,  and  the  second 
as  the  conjugate,  liyperbola. 

The  foci  of  the  conjugate  hyperbola  are  on  the  J'-axis,  and, 
since  c  =  Va^  +  h"^  is  the  same  for  each  hyperbola,  the  four  foci  lie 
on  a  circle  of  radius  c  and  center  at  the  origin  (Fig.  84). 

The  eccentricity  of  the  conjugate  hyperbola  differs  from  the 

eccentricity  of  the  primary  hyperbola.     The  former  is  -  and  the 

h 

latter  is  -• 
a 

The  asymptotes  of  the  conjugate  hyperbola  coincide  with  the 

asymptotes  of  the  primary  hyperbola.     For,  from  the  equation  of 

the  conjugate  hyperbola,  we  have 


^  =  ±-V.^•2  + 


Hence,  as  x  increases  indefinitely,  the  curve  approaches  nearer 
and  nearer  to  the  lines  y 
of  the  primary  hyperbola. 


and  nearer  to  the  lines  y  =  ±-^'     But  these  are  the  asymptotes 


EXERCISES 

1.  Show  that  the  foot  of  the  perpendicular  from  a  focus  of  an  hyperbola 
on  either  asymptote  is  at  a  distance  a  from  the  center  and  h  from  the  focus. 

2.  Show  that  the  circle  of  radius,  h,  whose  center  is  at  a  focus  of  an 
hyperbola,  is  tangent  to  the  asymptotes  at  the  points  where  they  cut  the 
corresponding  directrix. 

3.  Show  that  the  product  of  the  perpendiculars  let  fall  from  any  point  of 
an  hyperbola  on  the  asymptotes  is  constant. 

4.  Write  the  equation  of  the  hyperbola  conjugate  to  9  x"^  —  ?/2  =  9,  and 
find  the  lengths  of  its  semiaxes,  its  eccentricity,  the  coordinates  of  its  foci, 
and  the  equations  of  its  directrices. 

5.  If  e  and  e\  are  the  eccentricities  of  two  conjugate  hyperbolas,  show 
that  — I =  1.     Also  ae  =  bei. 

6.  What  is  the  eccentricity  of  the  equilateral  hyperbola  ?  Of  the  con- 
jugate to  the  equilateral  hyperbola  ? 

108.    The  system  of  concentric  hyperbolas.     The  equation. 


136 


LOCI   OP   SECOND   ORDER 


[Chap.  VII. 


where  Ji  is  a  variable  parameter,  is  the  equation  of  a  system  of 
concentric  hyperbolas  (Fig.  85).  All  the  hyperbolas  contained 
in  the  system  have  the  same  asymptotes,  as  can  be  shown  by 
solving  (1)  for  y  and  then  allowing  x  to  increase  indefinitely.  If 
A;  is  a  negative  number,  (1)  is  the  equation  of  a  conjugate  hyper- 
bola. As  A'  increases  in  value,  the  hyperbola  approaches  the 
asymptotes  closer  and  closer,  and  coincides  with  them  when  Jc  is 


Fig.  85 

zero  (Art.  106).  When  k  is  an  increasing  positive  number,  the 
hyperbolas  are  all  primary  and  recede  farther  and  farther  from 
the  asymptotes. 

The  contour  lines  about  two  contiguous  mountain  peaks  form  a 
rough  approximation  to  such  a  system  of  hyperbolas. 


109.    The  system  of  confocal  conies. 

foci  are  called  confocal.     The  equation 


Conies  having  the  same 


+ 


y 


a'-k     b'^-k 


1, 


(1) 


where  A;  is  a  variable  parameter,  is  the  equation  of  a  system  of  con- 
focal conies  (Fig.  86).  For,  suppose  a  >  b,  then  for  every  value  of 
k<i  ¥,  (1)  is  the  equation  of  an  ellipse.  The  distance  from  center 
to  focus  is  the  same  for  all  these  ellipses,  since 


c  =  V(a2  -  A^)  -  (62  _  A;)  =  Va^  -  b\ 


Art.  109] 


SYSTEM   OF  CONFOCAL   CONICS 


137 


As  k  approaches  b^,  the  semiconjugate  axis  of  the  ellipse  ap- 
proaches zero  and  the  semitransverse  axis  approaches  Va^  —  b^, 


Fig.  Sd 


and  therefore  the  ellipse  shrinks  to  the  segment  of  the  X-axis 
contained  between  the  foci. 

If  b^<k<.  a^,  (1)  is  the  equation  of  an  hyperbola  whose  semi- 
axes  are  Vct'  —  k  and  -\/k  —  b^.  The  distance  from  center  to  focus 
is,  in  this  case, 


c  =  V(a2  -  k)  -h  (fc  -  b')  =  Va^  -  b\ 

and  therefore  these  hyperbolas  are  confocal  with  the  ellipses. 

For  any  value  of  A;  >  a^,  equation  (1)  is  satisfied  for  no  'real 
values  of  x  and  y.  In  this  case,  (1)  is  said  to  be  the  equation  of 
an  imaginary  ellipse. 

Each  hyperbola  of  the  system  cuts  every  ellipse  at  right  angles 
and  vice  versa  (Exercises,  Art.  100). 

If  two  sets  of  curves  are  so  related  that  each  curve  of  either  set 
intersects  all  the  curves  of  the  other  set  at  right  angles,  the  two 
sets  of  curves  are  said  to  be  orthogonal  to  each  other.  Thus  the 
hyperbolas  and  ellipses  of  the  system  of  confocal  conies  form  two 
sets   of   curves   orthogonal   to   each  other.     Sets   of   orthogonal 


138  LOCI   OF  SECOND   ORDER  [Chap.  VII. 

curves  are  of  great  importance  in  mathematical  physics,  since 
they  represent  fields  of  force. 

EXERCISES 

1.  What  are  the  equations  of  the  two  conies  of  the  system 

(9  _  A)      (4  -  ^•) 
which  pass  through  the  point  {^2L^,  X^\  ? 

2.  Show  that  the  equation   —  +  ^  =  ^-  is  the  equation  of  a  system  of  con- 

centric  ellipses,  k  being  a  variable  parameter.  Discuss  the  equation  for 
various  values  of  k. 

3.  Show  that  the  equation  y'^  =  4  k(x  +  k)  is  the  equation  of  a  system  of 
confocal  parabolas.     Discuss  the  equation  for  various  values  of  A;. 

4.  What  system  of  conies  is  given  by  the  equation  —  +  ^  =  1,  k  being  a 

variable  parameter  ?  Show  that  the  x-intercept  of  a  tangent  to  any  one  of 
these  conies  is  independent  of  k.  How  can  this  fact  be  utilized  to  construct 
the  tangent  at  any  point  on  one  of  the  conies  of  the  system  ? 

5.  Discuss  the  system  of  circles  given  by  the  equation  x^  +  ?/2  —  ffl2  _  2  A;y  =  0, 
k  being  a  variable  parameter. 

6.  Discuss  the  system  of  circles  given  by  the  equation  x2  +  2/2_^(5[2_ 2  mx=0, 
m  being  a  variable  parameter. 

7.  Show  that  the  two  systems  of  circles  in  exercises  5  and  6  form  two  sets 
of  circles  orthogonal  to  each  other.     Draw  a  figure  illustrating  this  exercise. 

MISCELLANEOUS  EXERCISES 

1.  Find  the  equations  of  the  tangents  to  the  ellipse  x^  +  4  y-  =  16  which 
pass  through  the  point  (2,  3). 

2.  Find  the  equations  of  the  tangents  to  the  hyperbola  2ofi  —  Zy'^  —  18 
which  pass  through  the  point  (4,  —  Vo). 

3.  Find  the  coordinates  of  the  points  of  contact  of  the  tangents  in  ex- 
ercises 1  and  2. 

4.  For  what  value  of   k   \s  y  =  2x  ■\-  k    a    tangent  to   the  hyperbola 
x2  -  4  2/2  :=  4  ? 

5.  For  what  value  of  m  isy  =  m:c  +  2  a  tangent  to  the  ellipse  x^  +  4  j/^  =  1  ? 


Art.  1091  SYSTEM   OF   CONFOCAL  CONICS  139 

6.  What  relation  connects  A,   B,   and    C,   if  Ax  +  By  +  C  =  0  is  a 
tangent  to  the  parabola  ?/-^  =  4  x  ? 

7.  Are   the   following   points   on,    inside,    or   outside    the    hyperbola 
4  x2  -  2/2  =  4  y     («)  (1,  3),  (5)  (2,  1),  (c)  (3.25,  3). 

8.  The   coordinates   of   one    extremity    of  a  diameter    of   the   ellipse 

—  +  ^  =  1  are  xi  —  a  cos  di  and  yi-h  sin  ^1.     Show  that  the  coordinates  of 
a-      &'^ 

one   extremity  of   the   conjugate   diameter   are    given    by   the    equations 

X2  =  —  a  sin  di  and  y2  —  h  cos  di. 

9.  Show  that  the  segments  of  any  line  contained  between  an  hyperbola 
and  its  asymptotes  are  equal  in  length. 

10.  Find  the  equation  of  the  tangent  to  the  parabola  ij"^  =  4:px  which  has 
equal  intercepts. 

11.  The  earth's  orbit  is  an  ellipse  whose  eccentricity  is  .01677  and  whose 
major  semiaxis  is  93  million  miles,  the  sun  being  at  one  focus.  Find  the 
greatest  and  the  least  distance  from  the  earth  to  the  sun. 

12.  Find  the  angle  which  one  diameter  of  an  ellipse  makes  with  its 
conjugate  diameter. 

13.  A  comet  moves  in  a  parabolic  orbit  with  the  sun  at  the  focus.  If  the 
comet  is  2  million  miles  from  the  sun  when  the  line  from  sun  to  comet  makes 
an  angle  of  60°  with  the  axis  of  the  orbit,  find  the  least  distance  from  sun  to 
comet. 

14.  Show  that  the  bisector  of  the  angle  formed  by  lines  joining  any  point 
of  an  equilateral  hyperbola  to  the  vertices  is  parallel  to  an  asymptote. 

15.  Find  the  equation  of  the  locus  of  the  mid-points  of  chords  drawn 
from  one  end  of  the  major  axis  of  an  ellipse. 

16.  Show  that  the  ordinate  of  the  intersection  of  any  two  tangents  to  the 
parabola  y'^  =  4px  is  the  arithmetic  mean  of  the  ordinates  of  the  points  of 
contact,  and  the  abscissa  is  the  geometric  mean  of  the  abscissas  of  the  points 
of  contact. 

17.  Find  the  equation  of  the  locus  of  the  intersection  of  two  tangents  to 
the  parabola  ?/-  =  4  px,  if  the  sum  of  the  slopes  of  the  tangents  is  constant. 

18.  Show  that  the  angle  formed  by  any  two  tangents  to  the  parabola  is 
half  the  angle  formed  by  the  focal  radii  to  the  points  of  contact. 

19.  Any  two  perpendicular  lines  are  drawn  from  the  vertex  of  a  parabola. 
Show  that  the  line  joining  their  other  points  of  intersection  with  the  parab- 
ola cuts  the  axis  at  a  fixed  point. 

20.  Show  that  the  tangents  to  the  parabola  at  the  extremities  of  any 
chord  intersect  on  the  diameter  bisectins;  the  chord. 


140  LOCI   OF  SECOND   ORDER  [Chap.  VII. 

21.  Show  that  the  eccentricity  of  an  hyperbola  is  equal  to  the  secant  of 
half  the  angle  between  the  asymptotes. 

22.  Show  that  the  tangents  at  the  vertices  of  an  hyperbola  intersect  the 
asymptotes  at  points  on  the  circle  about  the  center  and  passing  through  the 
foci. 

23.  Show  that  the  product  of  the  distances  from  the  center  of  an  hyper- 
bola to  the  intersections  of  any  tangent  with  the  asymptotes  is  constant. 


CHAPTER   Vlir 

LOCI  OF  THE  SECOND  ORDER  EQUATIONS  NOT  IN  STANDARD 

FORM 

110.  Translation  of  the  coordinate  axes.  If  the  coordinate  axes 
are  translated  by  means  of  equations  (1),  Art.  77,  and  then  the 
primes  are  dropped,  the  standard  forms  of  the  equations  of  the 
several  conies  become : 

Circle:  (x  +  Ti)"-^  +  (// +  A;)^  =  r^,  (1) 

Ellipse:  (^±^+(^±_M^'  =  l,  (2) 

Primary  hyperbola :  (^±^  _  UL^  =  i,  (3) 

Conjugate  hyperbola  :  (^±^  _  iJlA^  =  _  1,  (4) 

Parabola,  X-axis  parallel  to  the  axis  of  the  curve : 

(y  +  hyi  =  4iJ(x  +  h),  (5) 

Parabola,  T^axis  parallel  to  the  axis  of  the  curve : 

(.r  +  /i)2  =  4i>(»/  +  A;),  (6) 

On  the  other  hand,  if  an  equation  of  the  second  degree  can  be 
reduced  to  one  or  the  other  of  the  above  forms,  the  locus  is  the 
corresponding  conic.  The  center  of  the  circle,  the  ellipse,  or  the 
hyperbola  is  then  the  point  (—  /*,  —  k),  and  the  vertex  of  the 
parabola  is  the  point  (—  h,  —  k),  Figs.  87  and  88. 

As  an  example,  take  the  equation 

9  x^  +  4  2/2  +  .54  X-  I6y  +  61  =0. 

Completing  the  squares  of  the  terms  in  x  and  the  terms  in  2/  separately,  the 

equation  can  he  written         f,,      ,   .^.„   ,    .,         -.xo      00 
^  9(x  +  S)^  +  4c{y  —  2)2  z=  36. 

Comparing  with  (2),  we  see  that  the  locus  is  an  ellipse  whose  center  is  the 
point  (—  3,  2)  and  whose  semiaxes  are  2  and  3. 

141 


142      EQUATIONS  NOT   IN   STANDARD  FORM     [Chap.  VIII.. 


If  any  one  of  the  equations  (1)  to  (6)  is  expanded  and  cleared 
of  fractions,  it  is  seen  to  be  a  special  case  of  the  equation 

rta?2  +  6t/2  +  2sr£c  +  2/2/  +  c  =  0,  (7) 

where  a,  b,  g,  f,  and  c  depend  upon  h  and  k.     We  are  led,  then,  to 
the  following 

Theorem.  The  equation  of  a  conic  referred  to  coordinate  axes 
parallel  to  the  axes  of  the  curve  (the  axis  and  tangent  at  vertex  in  case 
of  the  parabola)  has  the  form  (7). 


Fig.  87 


Fig.  88 


111.    Discussion  of  the  equation  ax^  +  by^  +  2gx  +  2fy  +  c  =  0. 

The  question  now  arises  :    Is 

ax'  +  by'-\-2yx  +  2fy  +  c  =  0  (1) 

the  equation  of  a  conic  for  any  given  set  of  values  of  the  coeffi- 
cients ?  We  shall  answer  this  question  by  a  discussion  of  the 
equation  (cf.  Art.  46). 

General  case.  Let  us  first  suppose  that  none  of  the  coefficients 
is  equal  to  zero,  and  we  may  farther  suppose  without  loss  of 
generality,  that  a  is  a  positive  number.  For,  if  a  is  a  negative 
number  in  any  particular  case,  we  can  change  the  signs  of  all  the 
terms  in  the  equation.     Equation  (1)  can  then  be  written  in  the 

f°^'^  -  No  /  ^\2  2  « 


a  x-\- 


+*(^'  +  5J='^+%-'- 


(2) 


Denote  the  right-hand  member  of  (2)  by  D,  then  the  nature  of  the 
locus  will  depend  upon  the  signs  of  b  and  D.  Thus,  if  b  is  nega- 
tive,  the  locus  is  an  hyperbola  which   is  primary  or  conjugate 


Art.  Ill]     DISCUSSION   OF  ax2  +  &2/2  + 2  ffx  + 2/2/ +  c=0  143 

according  as  D  is  positive  or  negative  (cf.  Art.  107).  If  h  and  Z) 
are  positive,  the  locus  is  an  ellipse.  There  is  no  locus  if  h  is 
positive  and  D  is  negative,  for  the  sum  of  two  positive  numbers 
can  never  be  negative.     The  locus  is  then  said  to  be  imaginary. 

If  either  a  or  6  is  zero,  the  above  method  fails.  But  if  a  is 
zero,  while  6  and  g  are  different  from  zero,  (1)  can  be  written 

and  if  &  is  zero,  while  a  and  /  are  different  from  zero,  (1)  can  be 
written 

In  either  case,  the  locus  is  a  parabola,  as  we  see  on  comparison 
with  equations  (5)  and  (6)  of  the  preceding  article. 

Special  cases.  If  D  is  zero  and  b  is  negative,  the  left-hand 
member  of  (2)  is  the  product  of  two  linear  expressions,  and  there- 
fore the  locus  of  (2),  and  consequently  the  locus  of  (1),  is  a  pair 
of  intersecting  straight  lines  (cf.  Art.  85).     If  b  is  positive,  the 

locus  consists  of  the  single  point,  (  —  -»  —  V  ),  since  this  is  the  only- 
point  whose  coordinates  will  then  satisfy  (2). 

Finally,  if  a  and  g,  or  b  and  /,  are  each  equal  to  zero,  equation  (1) 
contains  but  one  of  the  variables.  For  example,  if  b  and  /  are 
each  equal  to  zero,  (1)  becomes 

ax''  -f  2  gx  +c  =  0.  (5) 

If  the  roots  of  (5)  are  real  and  distinct,  the  locus  consists  of  a 
pair  of  lines  parallel  to  the  F-axis.  If  the  roots  of  (5)  are  equal, 
the  locus  consists  of  a  single  line  parallel  to  the  y-axis.  If  the 
roots  of  (5)  are  imaginary,  there  is  no  locus,  but  we  shall  say,  in 
this  case,  that  the  locus  consists  of  a  pair  of  imaginary  lines.  In 
these  special  cases,  the  locus  is  said  to  be  degenerate. 

We  shall  find  it  convenient  to  say  that  the  locus  of  (1)  is  a  conic, 
but  that  in  certain  cases,  the  conic  is  imaginary,  or  degenerates  into 
a  single  line,  or  into  a  pair  of  real  or  imaginary  lines,  or  consists  of 
a  single  point. 


144      EQUATIONS  NOT    IN   STANDARD   FORM    [Chap.  VIII. 

As  an  example  of  the  foregoing  analysis,  consider  the  equation 
9  x^  -  16  y-^  -  30  X  +  9(3  ?/  -  108  =  0. 

Here  we  find  that  a  is  positive,  D  is  zero,  and  b  is  negative.  Therefore  the 
locus  consists  of  two  intersecting  lines.     The  equations  of  these  lines  are 

3(x-2)i4(j/-3)  =  0, 

and  they  intersect  in  the  point  (2,  3). 

As  a  second  example,  consider  the  equation 

9  x2  +  2  ?/ -  18  X  +  8  ?/ +  17  =  0. 

In  this  case,  a  is  positive,  D  is  zero,  and  h  is  positive.  Hence  the  locus  con- 
sists of  a  single  point.  Completing  the  squares  of  the  terms  in  x  and  the 
terms  in  y  separately,  the  equation  becomes 

9(x-l)2  +  2(?/  + 2)-^  =  0. 

Therefore  the  point  (1,  —  2)  is  the  only  point  whose  coordinates  satisfy  the 
given  equation. 

A  summary  of  the  possible  loci  of  the  equation 
ax^  +  hi/  +  2  gx  +  2/y  +  c  =  0 

is  given  in  the  following  table,  where  D  =  -^  -{--^ c. 

a       b 


a  ^  0,  5^0 

a=0  (or  &  =  0) 

a>0 
b>0 

ffl>0 
6<0 

a  =  0,g^O,b^O 
(or  6  =  0,  /^O,  rt  9^0) 

a  =  g  =  0,  b=^0- 
(or  &=/=0,  a^O) 

D>0 

Ellipse 

D>0 

Hyperbola 
(priraary) 

Parabolas 

A  pair  of  real  par- 
allel   lines,    a    single 
line,  or  a  pair  of  imag- 
inary lines ;  according 
as  the  roots  of  by^  + 
2fy  +  c  =  0    (or    a.x2 
+  2  gj-  +  c  =  0)     are 

D  =  0 
Point 

Z>  =  0 

Intersecting 
lines 

D      0 

Imaginary 

D<0 

Hyperbola 
(conjugate) 

real      and      distinct, 
equal,  or  imaginary. 

In  the  following  exercises,  use  is  to  be  made  of  this  table. 


Art.    112]         EQUATION   OF   SECOND   DEGREE  145 

EXERCISES 

1.  Determine  the  nature  of  the  loci  of  the  following  equations.  Find  the 
coordinates  of  the  center  and  the  coordinates  of  the  foci  of  each  ellipse  or 
hyperbola  ;  the  coordinates  of  the  vertex  and  the  coordinates  of  the  focus  of 
each  parabola.     Make  a  sketch  of  each  curve. 

(a)  2x^  +  Sy^-6x  +  ^y  =  10.  (b)  x^ +  2y^  -  6 x  +  ij  =  10. 

(c)   ix^-Sy^-4:X  +  8  =  0.  (d)  .r^  +  4x  -  2  ?/ =  15. 

(e)   3x^  -y2  +  0y^0.  (/)  y^  +  2x-'iy  =  7. 

2.  Determine  the  nature  of  the  loci  of  the  follow^ing  equations  : 

(a)  x^  +  y^  —  4 X  -  6 y  +  IS  =  0.  (b)   x-  —  y^  —  4 x  +  6  y  +  5  =  0. 

(c)   x2  -  5  r.  +  6  =  0.  (d)  y^  -  6  ?/  +  9  =  0. 

(e)   ?/2  -  6  ?/  +  10  =  0.  (/)  y2  _  6  2/  +  8  =  0. 

112.    The  general  equation  of  second  degree.     The  equation 

ax'  +  2  hxu  +  hii-  +  2  gx  +  2./)/  +  c  =  0  (1) 

is  called  the  general  equation  of  second  degree,  because  it  contains 
every  term  that  can  appear  in  an  equation  of  the  second  degree. 
We  will  now  prove  the  following  theorem. 

Theorem.  The  term  in  xy  can  be  removed  from  the  general 
equation  of  second  deg)-ee  hy  rotating  the  axes  through  a  positive 
angle  0,  less  than  90°. 

Keplacing  x  and  y  in  (1)  by  their  values  in  terms  of  x'  and  y' ; 

namely,  ,         „        ,    •     n 

''  X  =  jc  cos  6  —  y  sm  6, 

y  =  x'  sin  0  +  y'  cos  d, 
(Art.  78),  we  obtain 

a'x'^  +  2  h'x'y'  +  b'y"  +  2  g'x'  +  2f'y'  +  c  =  0,  (2) 

a'  =  a  cos2  ^  +  2  /i  sin  ^  cos  6>  +  6  sin^  0,  (3) 

b'  =  a  sin2  e-2h  sin  ^  cos  ^  +  6  cos^  0,  (4) 
2  h'  =  2  h{cos^  6  -  sin2  d)  -  2(a  -  b)  sin  0  cos  0 

^2h  cos  2  6-{a-b)  sin  2  6,  (5) 

2(7'  =  2f/cos^  +  2/sin^,  (6) 

2/'  =  2/ cos  ^  -  2  ^  sin  ^.  (7) 

If,  now,  we  can  choose  0  so  that  h'  shall  equal  zero,  the  term  in 
x'y'  will  drop  out  of  (2)  and  the  general  equation  will  be  reduced 
to  the  form     ^^,^,  ^  ^,^,,  ^  ^  ^,^,  ^  ^_^,^,  ^  ^^  ^  ^_  ^^^ 


where 


146      EQUATIONS  NOT   IN   STANDARD   FORM     [Chap.  VIII. 

Putting  h'  equal  to  zero  in  (5),  we  have  for  the  determination  of  6, 

2  h  cos  2  ^  -(a  -  6)  sin  2^  =  0, 

from  which 

tan  20  =  ^^.  (9) 

Since  the  tangent  of  an  angle  assumes  all  possible  positive  and 
negative  values  as  the  angle  increases  from  0°  to  180°,  it  follows 
that  it  is  always  possible  to  find  an  angle  6,  less  than  90°,  ivhicJi 
iviJl  satisfy  equatiori  (9).  If  the  axes  are  rotated  through  this 
angle,  the  term  in  x'y'  drops  out  and  the  general  equation  is  thus 
reduced  to  the  form  (8). 

Equation  (8)  has  the  same  form  as  that  discussed  in  the  pre- 
ceding article.  Therefore  we  can  say  that  the  locus  of  the  general 
equation  of  second  degree  is  a  conic,  hut  that  this  conic  may  be 
imaginary,  or  may  .consist  of  a  single  line,  or  of  a  pair  of  real  or 
imaginary  lines,  or  of  a  single  2'>oint. 

The  values  of  the  coefficients  a'  and  b'  can  be  found  easily 
from  equations  (3),  (4),  and  (5).     Thu«,  adding  (3)  and  (4),  we 

^^^^®  a'  +  h'  =a  +  h,  (10) 

and  subtracting  (4)  from  (3)  gives 

a'  -b'  =  2  h  sin  2  ^  +  (a  -  b)  cos  2  9.  (11) 

Squaring  (5)  and  (11)  and  then  adding,  we  obtain 

4  /i'2  +  (a'  -  &')2  =  4  ¥  +  (a  -  bf.  (12) 

Subtracting  (12)  from  the  square  of  (10)  gives 

a'b'  -h'^  =  ab-h-.  (13) 

When  the  coordinate  axes  have  been  rotated  through  the  angle 
given  by  (9),  we  have  seen  that  h'  becomes  zero.  Hence  equa- 
tions (10)  and  (13)  give  respectively  the  sum  and  product  of  the 
required  coefficients.  These  coefficients  are  then  the  roots  of  the 
quadratic  equation 

X^  -  («  +  b)\  +  ab-  7t-  =  0.  (14) 

The  roots  of  this  equation  are  always  real,  since  the  discriminant, 
(a  +  by  —  4(a6  —  h"^)  =  (a  —  6)-  +  4  h"^,  is  always  positive. 


Art.  112]  EQUATION   OP   SECOND   DEGREE 


147 


In  order  to  decide  which  of  the  roots  of  (14)  to  take  for  a', 
eliminate  cos  2  6  between  (9)  and  (11),  thus  obtaining 


2  /i(a'  -  6')  =  [4  W  +  (a  -  hf]  sin  2  6. 


(15) 


For  0  <  90°,  sin  2  0  is  positive.  Tlierefore  a'  must  be  so  chosen 
that  a'  —  b'  will  have  the  same  sign  as  h. 

The  roots  of  (14)  will  be  both  different  from  zero  if  ab  —  h^  =fc  0, 
and  will  be  alike  in  sign,  or  unlike  in  sign,  according  as  their 
product  ab  —  h^  is  positive,  or  negative. 

The  values  of  the  coefficients  g'  and  /'  can  be  computed  from 
equations  (6)  and  (7),  but  the  computation  is  often  tedious,  and 
can  be  avoided  frequently  by  a  translation  of  the  axes  (Art.  113). 
If  a  given  equation  of  the  second  degree  contains  no  terms  of  the 
first  degree ;  that  is,  if  g  and  /  are  each  equal  to  zero,  then,  by 
(6)  and  (7),  g'  and/'  are  also  each  equal  to  zero  and  the  foregoing 
analysis  serves  to  determine  the  nature  and  the  position  of  the 
locus. 

For  example,  consider  the  equation 

X-  +  2  xy  +  2  y'^  —  4  =  0. 

Here  tan  20  =  -^-^  =  -^—  =  -  2,  from  which  we  get  8  =  58°  17',  nearly, 
a  —  b      1  —  2 

If  the  axes  are  rotated  through  the  angle  8,  the  term  in  x'y'  will  drop  out. 

The  coefficients  of  x'^^  and  ?/'''  are  the  roots  of  (14)  which  becomes,  in  this 

case, 


The  roots  are  ^+^^  and  ^  -  V5 


Since 


2  2 

h  is  positive,  a'  —  b'  must  be  positive,  and 
therefore  we  choose 


h' 


—  — '—       and  b'  =  — 


V5 


2  2 

The  given  equation,  thus,  reduces  to 


3  +  V^.o  ,  3  -  VS    ,.,       . 
~~2 — 2 —       ^    '  ^^ 


2(3 -V5)      2(3+\/y) 


The  locus  is,  therefore,  an  ellipse  whose  semiaxes  are  V2(3  —  VS)  and 
V 2(3  +VE).    The  major  axis  coincides  with  the  new  F-axis  (Fig.  89). 


148      EQUATIONS  NOT   IN  STANDARD  FORM    [Chap.  VIII. 

EXERCISES 

1.  Determine  the  nature  of  the  locus  of  the  equation  5 X'+2xy  +  5y^  =  12. 
Find  the  angle  through  which  the  coordinate  axes  must  be  rotated  in  order 
to  remove  the  term  in  xy.  Plot  the  curve  and  both  sets  of  axes.  Find  the 
eccentricity  of  the  curve. 

2.  What  is  the  locus  of  each  of  the  following  equations  ? 

(a)  3x^-2xy  +  y-^-6  =  0.  (b)  Sx^  -  2xy +  y^  =  -Q. 

(c)  Sx^-2xy  +  if  =  0.  (d)  9x^-20xy  +  11  y^  -  [>0  =  0. 

(e)  25  a;2  _  60  xy  +  36  2/2  -  81  =  0. 

3.  Reduce  the  following  equations  to  standard  form.  Draw  the  figure 
for  each  exercise. 

(a)  x2  +  xj/  +  2/2  _  1  =  0.  (6)  x2  +  3  x?/  -  3  2/2  -  4  =  0. 

(c)  2  x2  -  12  xy  -  3  2/2  +  14  =  0.  (d)  43  x2  +  30  xy  +  59  y^  -  68  =  0. 

(e)  8  x2  -  12  x?/  +  3  2/2  -  9  =  0. 

4.  The  locus  of  the  equation  Ax"^  +  4:xy  +  y^  +  k  =0  is  two  straight 
lines  for  any  value  of  k.     Discuss  the  change  in  these  lines  as  k  varies. 

5.  Show  that  3  x2  +  2  hxy  +  12  2/2  =  3  is  the  equation  of  a  system  of  concen- 
tric conies,  h  being  a  variable  parameter.  Discuss  the  change  in  the  locus 
as  h  varies  from  a  great  negative  number  to  a  great  positive  number. 

113.  Removal  of  the  terms  of  the  first  degree.  If  the  terms  of 
the  first  degree  can  be  removed  from  the  general  equation  of 
second  degree,  this  can  be  done  by  translating  the  axes,  as  in 
Art.  79.  Let  m  and  n  be  the  coordinates  of  the  new  origin. 
The  equations  for  translating  the  axes  are  then 

x  =  x'  -{-  m  and  y  =  y'  -\-  n     (Art.  77). 

Substituting  in  the  general  equation,  (1),  Art.  112,  and  arranging 
according  to  the  powers  of  x'  and  y',  the  resulting  equation  can 

be  written  r     ,  .  ,    ,■. 

{am  +  an  +  g)X 

ax'^+21ix'y'  -\-hy''  +  2\ 

[  +  qim  +  hn^f)y'\ 

(m{am  +  lin  +  9')  1 
+  n{hm-\-hn+f)\=0.  (1) 

-\-{gm-irfn+c)       \ 

If  the  terms  of  first  degree  can  be  removed,  m  and  n  must  sat- 
isfy simultaneously  the  two  equations 

am,  +  Tin  -\-  g  -d,  /9\ 

hm  +  hn  +  f  =  0.  ^^' 


Arts.  113,  114]  FIRST   CASE,   06  - /i^  ^  0  149 


The  values  of  m  and  n  derived  from  these  equations  are, 

ah  —  Ji 


,n  =  M^zAu^ 


(3) 
ab  —  hr 


^^  hg  -af 


We  now  have  three  cases  to  consider  (cf.  Art.  83)  : 

1.  Equations  (2)  are  consistent  and  have  but  one  common 
solution  if,  and  only  if,  ab  —  h^  is  not  equal  to  zero.  For  then 
equations  (3)  give  but  a  single  pair  of  values  for  m.  and  n. 

2.  Equations  (2)  are  inconsistent,  and  therefore  have  no  com- 
mon solution,  if  ab  —  h'^  =  0  and  hf—  hg  ^  0.  For  then  hg—af^O 
and  the  numerators  in  (3)  do  not  vanish  while  the  denominator  is 
equal  to  zero. 

3.  Equations  (2)  are  dependent  and  therefore  have  an  indefi- 
nite number  of  common  solutions  if  ab  —  h^  =  0  and  hf—  bg  =  0. 
For  then  hg  —  af=  0  and  both  the  numerator  and  the  denomina- 
tor of  each  fraction  in  (3)  vanishes  and  any  pair  of  values  of  m  and 
n  that  satisfies  one  of  the  equations  (2)  also  satisfies  the  other. 

We  shall  consider  the  three  cases  separately,  and  we  assume 
that  in  no  case  is  h  equal  to  zero.  For,  if  h  is  equal  to  zero,  the 
general  equation  has  the  form  discussed  in  Art.  111. 

114.  First  case,  ah  —  h^  ^0.  Central  conies.  In  this  case  the 
terms  of  first  degree  can  be  removed.  Setting  the  values  of  m 
and  n  from  (3),  Art.  113,  in  (1)  and  dropping  the  primes,  (1) 
becomes 

ax^^2hxy  +  bf  +  yMllM.,&lj:i^  +  c  =  Q.         (1) 
ab  —  7r  ab  —  /r 

The  absolute  term  in  this  equation  is 

dhc  +  2fgh  —hg^  —  af^  —  ch^  ,c,  ^ 

ab-W  ^"^ 

For  simplicity,  let  A  represent  the  numerator  in  C2)  and  C,  the 
denominator.     Equation  (1)  can  then  be  written 

ax^  +  2  hocy  +  by^  +^  =  0.  (3) 


150      EQUATIONS  NOT   IN   STANDARD   FORM     [Chap.  VIII. 
A  cau  be  expressed  as  a  determinant.     Thus, 

(4) 


a    h 

ff 

h    b 

f 

ff    f 

c 

C  is  the  cofactor  corresponding  to  the  element  c,  and  the 
numerators  in  (3),  Art.  113,  are  respectively  the  cofactors  corre- 
sponding to  the  elements  g  and  /.  The  determinant  A  is  called 
the  discriminant  of  the  general  equation  of  the  second  degree. 

If  the  axes  are  now  rotated  so  as  to  remove  the  term  in  xy,  (3) 
becomes 

aV^'+6y^  +  ^=0,  (5) 

where  a'  and  b'  are  the  roots  of  (14),  Art.  112. 

If  ab  —  h^  >  0,  a'  and  b'  are  alike  in  sign  (Art.  112)  and  (5)  can 

be  written  „         , ,    ,„        ,  /^^ 

\a'\x'^+\b'\y'^  =  c',  (6) 

where  1  a'  \  and  |  b'  \  are  positive  numbers  and  c'  is   ±  — .     The 

locus  is  then  an  ellipse  which  is  real  or  imaginary  according  as  c' 
is  positive  or  negative. 

If  ab  —  li^  <  0,  a'  and  6'  are  unlike  in  sign  and  then  (5)  can  be 

''''■'^*®''  \a'\x"'-\b'\y''  =  c'.  (7) 

The  locus  is  then  an  hyperbola  which  is  primary  or  conjugate 
with  respect  to  the  axes  X',  Y'  according  as  c'  is  positive  or 
negative. 

Degenerate  conies.  If  A  =  0,  then  c'  is  zero,  and  the  locus  of 
(6)  is  a  single  point ;  namely,  the  origin,  and  the  locus  of  (7)  is  a 
pair  of  straight  lines  intersecting  at  the  origin. 

Neither  a'  nor  b'  can-  equal  zero,  since  a'  ■  b'  =  ab  —  li^  ^  0.  We 
conclude,  therefore,  that  in  this  first  case,  the  locus  of  the  general 
equation  of  second  degree  is  either  an  ellipse,  real  or  imaginary ; 
an  hyperbola ;  a  pair  of  real  and  intersecting  lines ;  or  a  single 
point.  But  the  lomis  is  never  a  parabola,  a  pair  of  parallel  lines, 
or  a  single  line. 

We  may  further  conclude,  from  equations  (3),  (6),  or  (7),  that 
the  locus,  whatever  it  may  be,  is  symmetrical  with  respect  to  the 


Art.  114] 


FIRST   CASE,   ab-h^^O 


151 


a    h 

9 

h    b 

f 

9    f 

c 

origin,  since  if  x^  and  y^  satisfy  any  one  of  these  equations,  —  x^ 
and  —  yi  will  also  satisfy  the  equation.  Therefore  the  locus  of 
the  general  equation,  in  this  first  case,  is  symmetrical  with  respect 
to  the  point  {m,  n).  Or,  in  other  Avords,  the  point  {m,  n)  is  the 
center  of  the  locus.  Hence  the  condition  ah  —  W^Q  characterizes 
the  central  conies. 

As  an  example  of  the  foregoing  analysis,  let  us  reduce  the  equation 
8:B2_^4x2/  +  5y2-|.8u;—  16  2/-16  =  0 

to  the  standard  form  and  thus  determine  the  nature  and  position  of  the 
locus. 

u.        (t       y 

Here  C  =  a6  -  /t^  =  40  -  4  =  36,  and  ^  ==   h     b     /"    =  -  1296. 

Also  from  (3),  Art.  113,  we  have 

m  =  —  1  and  n  =  2. 

Therefore  the  center  of  the  conic  is  the  point  (—  1,  2).  Again,  from  (9), 
A^t.  112,  t^n  2^  =  1, 

from  which  we  find  d  =  26°  34',  nearly.  Hence  we  conclude  that  when  the 
axes  are  translated  so  that  the  new  origin  is  the  point  (—  1,  2),  the  given 
equation  wUl  take  the  form  (3),  or 

8x--]-ixy  +  6y^  —  SG  =  0, 

and  when  the  axes  are  rotated  through  the 
angle  26^^  34',  the  equation  will  take  the 
form  (5),  where  «'  and  b'  are  the  roots  of 

X2_  13X  +  36  =  0; 

i.e.  9  and  4.  Since  h  is  positive,  we 
choose  a'  =  9  and  b'  —  4.  The  given  equa- 
tion then  becomes 

a;2  ,  ?/2  _  . 


9  x2  +  4  ?/2  -  36  =  0 


or  -  +  i^  =  1. 
4       9 


Fig.  iK) 


The  locus  is  therefore  an  ellipse  whose  semiaxes  are  2  and  3      Figure  90 
shows  the  curve  and  the  three  sets  of  axes. 
As  a  second  example  consider  the  equation 

2  a;2  -  x?/  -  3  2/2  -  2  a:  +  18  2/  -  24  =  0. 

Here  we  find  A  =  0  and  C  =  -  ^-f.  The  roots  of  (14),  Art.  112,  are  therefore 
unlike  in  sign,  and  the  locus  consists  of  two  intersecting  lines.  The  left-hand 
member  of  the  given  equation  must  be  the  product  of  two  linear  factors 


152      EQUATIONS  NOT   IN   STANDARD   FORM     [Chap.  VIII. 


Fig.  91 


(Art.  85) .  Solving  the  given  equation  for  one 
of  the  variables,  considering  the  other  as  a  knovni 
number,  reveals  the  factors  at  once.  Thus,  solv- 
ing for  X,  we  have 

^^2M:_2      5y-U_ 
4  4 

Hence  the  locus  consists  of  the  two  lines 
2x-Sy  +  6  =  0  and  »•  +  ?/  -  4  =  0     (Fig.  91). 


EXERCISES 

1.  Reduce  the  following  equations  to  standard  form.  Determine  the 
coordinates  of  the  center  and  the  angle  through  which  the  axes  must  be 
rotated  in  order  to  remove  the  term  in  xy : 

(ffl)  5  x2  +  4  X2/  —  ?/"2  +  24  X  —  6  ?/  —  5  =  0.  (ft)  xy  +  y-  -}-  y  +  1  =  0. 

(c)  4:  xy  +  4: y'^  —  2 X  +  3  -  0.  (d)  x^  +  xy  +y^  +  3y  =:  0. 

(e)  x^  -2  xy  +  5y'^  -8y  =  0.  (f)x^  +  2xy  +  9y^  =  0. 

{g)  2  x^  -  Q  xy  +  6  y^  +  6  X  —  12  y  +  9  =  0. 

115.  Second  case,  ab  -  h'^  =0  and  hf-bg=p  0.  Non-central  con- 
ies. In  this  case  the  terms  of  first  degree  cannot  be  removed, 
since  equations  (2),  Art.  113,  are  inconsistent  and  have  no  com- 
mon solution.  We  begin  the  discussion,  therefore,  by  rotating 
the  axes  so  as  to  remove  the  term  in  xy.  The  angle  through 
which  the  axes  must  be  rotated  is  determined  from  (9),  Art.  112. 
After  rotation,  the  general  equation  assumes  the  form  (8),  Art. 
112,  where  a'  and  b'  are  the  roots  of  (14)  and  g'  and/'  are  deter- 
mined from  (6)  and  (7).  But  in  the  case  we  are  discussing, 
ah  —  /i^  =  0  and  equation  (14)  becomes 

X"  -  (a -\- b)X=  0 

whose  roots  are  0  and  a  -\-  b.  We  may  assume  that  a  is  positive 
(Art.  Ill)  then  b  is  also  positive,  since  the  product  ab  is  positive 
(  =  /i^).  Hence  we  choose  a'  =  a  4-  b,  or  b'  =  a  +  b,  according  as 
h  is  positive  or  negative  (Art.  112).  The  general  equation  is  then 
reducible  to  one  or  the  other  of  the  forms 


(a  +  b)3c''^  +  2  g'xJ  +  2  fy'  +  c  =  0, 
or  (a  +  6)2/'2+2sr'a?'  +  2/'*/'  +  c  =  0, 

according  as  li  is  positive  or  negative. 


(1) 
(2) 


Art.  115]     SECOND   CASE,    06-/1^  =  0  AND  hf  -  bg  ^  0 


153 


We  must  now  determine  g'  and/'.     From  (9),  Art.  112,  we  have 

,      0/1        2tan^  2/t 

tan  2  6  = = 

1  —  tan^  0     a  —  b 

Solving  this  quadratic  for  tan  6,  we  obtain 


tan  0^ — -  ± 


H 


by 


h^ 


+  1. 


But  since  h^  =  ab,  the  expression  under  the  radical  is  a  perfect 

square.     Therefore,  tan  6  =  -  or  —  -,  from  Avhich  sin  6  and  cos  9, 

h  h 

and  thence  g'  and  /'  can  be  calculated.     But,  since  0  <  90°,  tan  0, 

sin  0,  and  cos  0  are  all  positive.     Hence  we  have  the  following 

results,  where  the  sign  before  the  radical  is  positive : 


h  positive 

h  negative 

tan  e  = 

b 
h 

a 
h 

sin  d  = 

b 

a 

Vb-^  +  /i2 

Va2  +  A2 

cos^  = 

h 

-h 

y/b-^  +  h^ 

Va^  +  h^ 

ff'  = 

hg  +  bf 

Vb-^  +  li^ 

af-  hg 

Vd^  +  h^ 

/'  = 

hf-bg 

-  (hf  +  ag) 

V62  +  ^2 

(3) 

(4) 


Since  neither  hf—  bg  nor  hg  —  af  is  zero,  we  see  that  if  h  is 
positive,  and  therefore  the  equation  reducible  to  the  form  (1),  /' 
cannot  equal  zero;  and  if  h  is  negative,  and  the  equation  re- 
ducible to  the  form  (2),  g'  cannot  equal  zero.  Hence,  on  com- 
parison with  equations  (3)  and  (4),  Art.  Ill,  we  conclude  that 
the  locus  of  the  general  equation,  in  this  case,  is  necessarily  a 
parabola. 


154      EQUATIONS  NOT   IN   STANDARD   FORM     [Chap.  VIII. 

As  an  example,  let  us  reduce  the  equation 

x2  —  2  x?/  +  2/2  —  2  2/  -  1  =  0 

to  the  standard  form  and  thus  determine  the  position  of  the  locus. 

Since  ah  —  fi^  is  here  equal  to  zero  and  lifis  not  equal  to  hg,  the  locus  is  a 
parabola.  Again,  since  h  is  negative,  we  choose  the  form  (2).  Computing 
g'  and/'  from  (3)  and  (4),  the  given  equation  becomes 

2  2/'2  -  V'2  x' -  ^2  ?/' -  1  =  0. 
Completing  the  sqnare  of  the  terms  in  ?/',  vre  have 

1 


y'-- 


2y/2 


V2 


■     4V2/ 


Comparing  with  (5),  Art.  110,  we 
see  that  the  vertex  of  the  parab- 
ola, referred  to  the  new  axes,  is 

the     point  ( ^,   ^^V      If 

V     4\/2     2V2/ 
the  axes  are  translated  so  that  this 
point  is  the  new  origin,  the  equa- 
tion reduces  to  the  standard  form 

9       v''2 

y  =-7-»^> 


Fig.  92  where    the    primes    have    been 

dropped. 

The  angle  through  which  the  axes  have  been  rotated  is  given  by  the 

equation 

tan  61=--  =1, 
h 

Therefore  d  =  45^.     Figure  92  shows  the  locus  and  the  three  sets  of  axes. 


EXERCISES 

1.  Reduce  the  following  equations  to  the  standard  form .  Determine  the 
angle  through  which  it  is  necessary  to  rotate  the  axes  in  order  to  remove  the 
term  in  xy^  and  the  coordinates  of  the  vertex  referred  to  the  original  axes  : 

(a)  x2  -  2  a;?/  -h  2/2  -  8  X  +  16  =  0. 
(6)  x2  —  2  x?/  +  2/^  +  2  X  —  2/  -  1  =  0. 

(c)  4  x2  +  4  X2/  -I-  2/^  —  4  X  =  0. 

(d)  9  x2  +  12  X2/ -h  4  2/2  -  2  2/ =  0. 

(e)  x2  -h  4  X2/  +  4  2/2  -  6  X  -I-  8  2/  -t-  1  =  0. 


Art.  116]    THIRD   CASE,   afc  - /i^  =  0  AND   hf  -  bg  =  0  155 

116.  Third  case,  ab—h^=0  and  hf—bg=0.  Since  ah  —  h^ 
is  again  eqnal  to  zero,  the  general  equation  is  reducible  to  the 
form  (1),  or  the  form  (2),  of  the  preceding  article,  according  as  h 
is  positive  or  negative.  But  here  hf—bg  =  0  and  af  —  hg=Q. 
Hence,  if  h  is  positive,  /'  =  0 ;  and  if  7i  is  negative,  g'  =  0. 
Consequently  equations  (1)  and  (2)  of  the  preceding  article  be- 
come respectively      .        ,  ^   ,.,      r.    ,  r 

^""^  (a  +  b)y"  +  2f'y'+c  =  0.  '  (2) 

Each  of  these  equations  contains  but  a  single  variable.  There- 
fore, in  this  case  (cf.  Art.  Ill),  the  locus  consists  of  a  pair  of 
parallel  lines;  a  single  line  ;  or  ci  pair  of  imaginary  lines,  according 
as  the  roots  of  the  equation,  (1)  or  (2),  are  real  and  distinct;  equal; 
or  imaginary. 

It  is  not  necessary  to  calculate  the  coefficients  g'  and  /'  in 
order  to  determine  the  nature  of  the  locus  of  an  equation  satisfy- 
ing the  conditions  of  this  third  case.  For,  since  a  is  not  zero, 
the  general  equation  of  the  second  degree  can  be  written 

arx-  +  2  ahxy  +  aby^  +  2  agx  +  2  afy  +  ac  =  0,  (3) 

and  since  ab  =  h-  and  af  —  hg,  (3)  becomes 

{ax  +  hyf  -H  2  g{ax  +  hy)  +  ac  =  0.  (4) 

The  locus  then  consists  of  a  pair  of  parallel  lines,  a  single  line, 
or  a  pair  of  imaginary  lines  according  as  g^  is  greater  than,  equal 
to,  or  less  than  ac.     For  example,  the  locus  of  the  equation 

4.T2-f  12a;?/  +  9/  +  4;^  +  6y  +  l  =0 
is  a  single  line,  since  here  g-  =  ac.     The  equation  of  the  line  is 

'^'^'^^  2x+3y  +  l  =  0. 

EXERCISES 

1.  Determine  the  nature  of  the  loci  of  the  following  equations.  Draw 
the  locus  when  possible. 

(«)  x^  —  2  xy  +  y^  +  2  tj  —  2  X  +  1  =  0. 
(6)  4  x^  +  12  xy  +  9  y^  +  4  X  +  6  y  +  2  =  0. 
(c)  x^  +  2xy  +  y'^—1  =  0. 
Id)  9  a;2  -  12  x?/  +  4  2/2  +  1.5  X  -  10  2/  +  6  =  0. 


156      EQUATIONS  NOT   IN  STANDARD   FORM     [Chap.  VIII. 

2.  If  a  =  0,  in  the  third  case,  show  that  the  general  equation  is  neces- 
sarily by'^  +  2fy  -\-  c  =  0,  and  therefore  the  locus  is  a  pair  of  parallel  lines,  a 
single  line,  or  a  pair  of  imaginary  lines  according  as  f'^  is  greater  than,  equal 
to,  or  less  than  be. 

117.    Recapitulation.     The  results  of  the  foregoing  three  arti- 
cles can  be  exhibited  in  tabular  form  as  follows : 
Loci  of  the  general  equation  of  the  second  degree 


A  = 


a     h 

y 

h     b 

f 

9    f 

c 

C=ab-h\ 


First  case, 
ab-h^^O 

Second  case, 
ab-h'^=0,  hf-bg=j^O 

Third  case, 
ab-h-^  =  0,  hf-bg  =  0 

(7>0 

C<0 

Parabolas 

Parallel     lines,     a 

A^^O 

Ellipse, 
real  or 
imag. 

Hyperbola 

single  line,  or  imag- 
inary lines 

A  =  0 

Point 

Intersecting 
lines 

EXERCISES 

1.  Analyze  the  following  equations.     What  is  the  locus  of  each  ? 
(a)  x^  +  Qxy +  y'^- 4  X- 12  y  +  10  =  0.    (6)  x^  -  xy  +  y^  +  3  x  =  0. 

(c)    9  ,x2  _  30  xy  -I-  25  ?/2  _  10  X  =  0.  (d)  2oi^  —  xy  +  bx -2y  -\- Q  =Q. 

2.  Analyze  each  of  the  following  equations  and  draw  the  corresponding 
locus. 

(a)  ,x2  -  2  x?/  -f-  ?/2  -  10  X  -  6  2/  +  25  =  0.   (6)  x^  -  x?/  -|-  5  x  -  2  ?/  -|-  6  =  0. 
(c)  2  x2  +  xy  -I-  J/''^  -  5  X  -  10  ?/  -f  18  =  0.    (fZ)  x2  -(-  3  xy  +  2  2/2  _  ^  _  2/  =  0. 

3.  The  locus  of  the  equation  3  x^  —  3  xy  —  y^  +  15  x  +  10  y  —  24  =  0  is  an 
hyperbola  ;  find  the  equations  of  its  asymptotes. 

Suggestion.  The  center  of  the  curve  is  found  to  be  the  point  (0,  5). 
The  standard  form  of  the  equation  is  3  x'-  —  7  y^  =  2.  Hence  the  equations  of 
the  asymptotes,   referred    to  the  axes   of    the  curve,    are  3  x^  —  7  y^  =  0 


Arts.  117,  118]  TANGENTS  157 

(Art.  106).     When  the  coordinate  axes  are  transformed  back  to  the  original 
position,  the  equations  of  the  asymptotes  become 

3  x2  -3x(2/  -  5)  -  {y-  by-  =  0, 

or  2/_5  =  (-3±V2"l)| 

4.  For  what  value  of  k  is  the  locus  of  x^  +  2  xj/  +  2  ?/2  +  ,t  +  ^•  =  0  a  pair 
of  straight  lines  ?    Are  these  lines  real  or  imaginary  ? 

5.  If  the  locus  of  the  general  equation  of  second  degree  in  x  and  «/  is  a 
central  conic,  show  that  the  equation  can  be  written  in  the  form 

a{x  —  mY  +  2  h{x  —  m)  {y  —  n)  +  h{y  —  n)-  = , 

where  to  and  n  are  the  coordinates  of  the  center,  and  A  and  C  have  the 
meanings  assigned  in  Art.  117. 

6.  Making  use  of  the  preceding  exercise,  show  that  the  equations  of  the 
asymptotes  of  any  hyperbola  are 

a(x  —  to)-  +  2  h,{x  —  m){y  —  n)  +  b{y  —  n)-  =  0. 

Apply  this  method  to  find  the  equations  of  the  asymptotes  of  the  hyperbola 

x2+6  X2/  +  2/2  -  4  X  —  12  ?/  -f-  10  =  0. 

7.  Find  the  coordinates  of  the  vertex,  the  coordinates  of  the  focus,  and 
the  equation  of  the  directrix  of  the  parabola 

x2_4x?/  +  4?/2-4x-22/  +  8  =  0. 

TANGENTS    AND   DIAMETERS 

118.  Tangents.  It  is  often  convenient  to  write  the  equation 
of  a  tangent  to  a  conic  at  a  given  point  without  first  having  to  re- 
duce the  equation  to  the  standard  form.  Suppose  the  equation 
has  the  general  form 

ax'  +  2  hxy  +  hy''  +  2gx  +  2fy +  c  =  0.  (1) 

Let  Pi{xi,  ?/i)  and  P-iix^,  y^)  he  any  two  points  on  the  curve.    Then 
we  must  have 

ax,^  +  2  hx,y,  +  hy,^  +  2  gx,  +  2fy,  +  c  =  0,  (2) 

axi  +  2  hx^y^  +  fty^'  +  2  gx^  +  2fy^  + c  =  0.  (3) 

Subtracting  (3)  from  (2),  we  obtain 

a{xy^  -  X.J)  +  2  li{x{y^  -  x,y^)  +  h{y^~  -  y.2^)+2  g{x,  -  x,) 

+  2f{y,-y,)  =  Q.  (4) 


158      EQUATIONS  NOT   IN   STANDARD   FORM     [Chap.  VIII. 

Dividing  (4)  by  x^  —  X2,  we  have 

a(x,  +  x,)  +  2h  ^'^y'  -  ^^^^  +  6  ^y^  +  y-^^^'\-  y^^  +  2g+2fy^^^^  =  0. 

(5) 
Now  -'^  ~  ■  -  is  the  slope  of  the  secant  PiPj.     Let  this  slope  be 
represented  by  m.     The  term  ^'"'{^  ~  ^'f'  can  be  written 
a^i.Vi  -  ^-1^/2  +  ^xVi  -  ^-iD-i 

«V1   ~~'  t^l/o 

and  is  therefore  equal  to  mx^  +  2/2.     Hence  (5)  becomes 

a{x^  +  0^2)  4-  2  hiinfix^  +  ^/j)  +  6(^1  +  y^'m  -\-2g^  2fm  =  0.     (6) 

Solving  for  m,  we  obtain 

^  ^  _  a(xi  +  a;^)  +  2  %2  +  2  .g  / ^s 

2^a;i  +  6(?/i  +  ^2)  +  2/  ^ 

Equations  (2)  to  (7)  hold  as  long  as  Pj  and  P2  are  on  the  curve. 
When  P2  approaches  Pj  along  the  curve,  the  secant  approaches 
the  position  of  the  tangent  at  Pi,  and  in  the  limit  coincides  with 
it  (Art.  97,  second  method).  Hence,  making  x^  =  x^  and  2/2  =  Ui  ir^ 
(7),  we  have  the  slope  of  the  tangent  at  Pj ;  that  is, 

ax    +hy   +(j 

m  =  —  — i i .  (8) 

hXj^  +  hij^  +f 

The  equation  of  the  tangent  at  P,  is,  therefore, 

ax,  +  hy,  -\-  a  X  . 

hx,  +  byi  +f 

Clearing  of  fractions  and  reducing  by  means  of  equation  (2),  we 
have 

ax,x  +  h  (x,y  +  y^x)  +  hy,y  +  g{x,  +  x)  +  f{y^  +  y)  +  c  =  0.     (9) 

This  equation  is  easily  remembered,  since  if  the  subscripts  are 
removed,  it  returns  to  the  original  form  (1). 

A  convenient  way  of  writing  (9)  is  the  following  : 

X  {ax^  +  %i  +  9') 
+  y(hx,  +  hy,+f)  (10) 

+  (^^i+./^i  +  c)  =  0. 


Arts.  118,  119]  DIAMETERS  159 

Either  (9)  or  (10)  is  the  equation  of  a  straight  line  whether  the 
point  Pi  is  on  the  conic  or  not.  If  P^  is  not  on  the  conic,  then 
(9)  or  (10)  is,  by  definition  (Art.  104),  the  equation  of  the  polar 
line  of  Pi  with  respect  to  the  conic  whose  equation  is  (1). 

EXERCISES 

1.  Find  the  equations  of  the  tangents  to  the  following,  at  the  points 
indicated. 

(a)  a;"-^  +  4  2/2  +  5  X  =  0,  at  the  points  whose  ordinate  is  1. 

(6)  xy  =  4,  at  the  point  whose  abscissa  is  2. 

(c)  a;2  _|_  xy  +  4  =  0,  at  the  point  whose  abscissa  is  2. 

{d)  y~  +  2  a-?/  —  3  =  0,  at  the  point  whose  ordinate  is  —  1. 

(e)  x^  —  3  xy  —  4  y^  +  9  =  0,  at  the  points  whose  ordinate  is  2. 

2.  Find  the  equation  of  the  polar  line  of  the  point  (1,  2)  with  respect  to 
the  conic  x^  —  3  xy  +  y-  =  4.     Draw  the  figiire  to  illustrate  the  problem, 

3.  In  exercise  7,  Art.  117,  show  that  the  directrix  is  the  polar  line  of  the 
focus  with  respect  to  the  given  laarabola. 

119.  Diameters.  In  case  of  a  central  conic,  we  have  found 
that  the  coordinates  of  the  center  are  the  values  of  m  and  n 
which  satisfy  equations  (2),  Art.  113.  This  amounts  to  saying 
that  the  center  of  the  conic  is  the  point  of  intersection  of  the 
lines  whose  equations  are 

hx  +  by+f  =  0.  ^  ^ 

Hence   these   two   lines   are  diameters  of   the  conic  (Art.  101). 
The  equation  of  any  diameter  is,  therefore  (Art.  84), 

(ax  +  hy  +g)+k  (Jix  +  by  +/)  =  0  (2) 

where  fc  is  a  variable  parameter. 

Let  P(a;i,  y^)  be  any  point  on  the  conic.  When  the  diameter 
(2)  passes  through  P,  A;  has  the  value 

o.i'i+7^?/i  +g 
hXi+by^-{-f 

But  this  is  the  slope  of  the  tangent  at  P,  (8),  Art.  118,  and  hence, 
also  the  slope  of  the  diameter  conjugate  to  (2),  Art.  102.     There- 


160      EQUATIONS  NOT   IN   STANDARD   FORM    [Chap.  VIII. 

fore,  the  parameter  k  is  the  slope  of  the  diameter  conjugate  to  (2). 
The  slope  of  (2)  is 

h  +  hk 
Hence,  k   and — —    are  the  slopes  of  a  pair  of  conjugate 

/I  — |-  ufC 

diameters  of  the  conic  tvhose  equation  has  the  general  form  (1)  of 
the  preceding  article. 

The  two  diameters  will  be  perpendicular  to  each  other,  and 
therefore  the   axes    (Art.    102,    exercises    12  and  13),  when  the 

product  of  their  slopes  is  —  1 ;  that  is,  when  — ~^ =  1,  or 

h  +  bk 

hk^  +{a-b)k-7i  =  0.  (3) 

If  ki  and  ^2  ^-i"©  the  roots  of  (3),  the  equations  of  the  axes  are 
(aoc  +  hy  +  g)+Ti.-^{hx  +  by  +  f)  =0,  . 

(aac  +  hij  +  g)+  Ji:..2(hx  +  by  +f)  =  0. 

The  roots  of  (3)  are  always  real,  since  the  discriminant, 

4  h^  +  (a  -  by, 
is  necessarily  positive. 

We  have  seen  (Art.  106)  that  an  asymptote  of  an  hyperbola  is 
a  self -con  jugate  diameter.  But  if  the  slope  of  any  diameter  of  a 
conic  is  equal  to  the  slope  of  its  conjugate  diameter,  we  must  have 

7.  __a-{-hk 
~~  h  +  bk' 
or  bk^  +  2hk  +  a  =  0.  (5) 

If  k'  and  k"  are  the  roots  of  (5),  the  equations  of  the  asymptotes 

(ax  +  hy +  g)+ Jc'ihac  +  by +f)=^Of  ,g 

and  (ax +  hy +  g)  +  k"(hx+ by  +  f)-0.  ^^ 

The  roots  of  (5)  are  real  and  unequal  only  if  a&  —  A^  <  0 ;  that  is, 
only  if  the  conic  is  an  hyperbola  or  a  pair  of  intersecting  lines 
(Art.  117). 

As  an  example,  consider  the  equation 

2  a-2  -f  4  x?/  —  2/2  +  4  a;  -  2  ?/  +  3  =  0. 
Here  equation  (8)  is  ^  ,,.2  ,  3  i-  _  9  —  0 


Art.  119]  DIAMETERS  161 

whose  roots  are  |  and  —  2.     Hence,  from  (4),  the  equations  of  the  axes  are 

2  X  +  y  +  I  =  0, 
and  X  —  2y  —  2  =  0. 

Equation  (5)  becomes  in  this  case 

]c-2-4k-2  =  0, 
whose  roots  are  2  ±  V6.     Hence  the  equations  of  the  asymptotes  are 

(2x  +  22/  +  2)  +  (2±V6)(2x-y-l)=0. 
The  student  should  draw  a  figure  illustrating  this  example. 

The  diameters  of  a  parabola  are  perpendicular  to  the  directrix 
(Art.  101)  and  therefore  perpendicular  to  the  tangent  to  the  curve 
at  the  vertex.  We  have  seen  (Art.  115)  that  the  equation  of  a 
parabola  is  reducible  to  one  or  the  other  of  two  forms  by  a  rotation 
of  the  axes  through  an  angle  ^<90°.  But  if  h  is  positive,  the 
new  X-axis  is  parallel  to  the  tangent  at  the  vertex  (Eq.  1, 
Art.  115) ;  and  if  h  is  negative,  the  new  X-axis  is  parallel  to  the 
axis  of  the  curve  (Eq.  2,  Art.  115).  Hence,  from  the  values  of 
tan  9  in  these  two  cases,  we  conclude  that  the  slope  of  a  diameter 

to  the  jyarabola  is or according  as  h  is  positive  or  negative. 

But  -  =  -  •     Consequently, is  the  slope  of  any  diameter. 

b      h  ^         b 

The  slope  of  the  tangent  to  the  parabola  at  the  point  P(x,  y)  is, 

(8),  Art.  118, 

m=      ax  +  hy-\-g 

hx  +  by+f 

This  tangent  is  perpendicular  to  the  diameters  of  the  curve  if 

—  =  1,  or  in  other  words,  if  the  coordinates  of  the  point  of  con- 
b 

tact  satisfy  the  equation 

(ax  +  hy  +  g)7i  ___^  ,j. 

{hx  +  by+f)b  ■  ^  ^ 

Clearing  of  fractions  and  remembering  that  h^  =  ab,  (7)  becomes 
(a  +  b)(hac  +  bij)+hg  +  fb  =  0.  (8) 


162      EQUATIONS   NOT   IN   STANDARD   FORM     [Chap.  VIII. 

Now  (8)  is  the  equation  of  the  axis  of  the  parabola.     For  it  is  a 
diameter  of  the  curve  since  it  has  the  slope 

0 
As  an  example,  consider  the  equation 

a:2  _  2  xy  +  y"-  —  2  ?/  -  1  =  0.     (cf .  Art.  115. ) 
From  (8),  we  get  tlie  equation  of  the  axis, 
2{x-y)+  1  =  0. 

If  we  solve  the  equation  of  the  curve  and  the  equation  of  the  axis  simulta- 
neously, we  get  the  coordinates  of  the  vertex.  In  this  example,  the  vertex  is 
the  point  ( —  |,  —  f )  •     The  equation  of  the  tangent  at  the  vertex  is,  therefore, 

y  +  I  ^_(.x  +  1),  or  X  +  ?/ +  f  =  0. 

As  a  second  example,  consider  the  equation 

9  a;2  +  24  xi/  +  16  2/2  —  62  X  +  14  2/  -  6  =  0. 

Here  we  find  the  equation  of  the  axis  is 

3x  +  4?/-2  =  0. 

Solving  simultaneously  with  the  given  equation,  we  find  that  the  vertex  is  the 
point  (.08,  .44).     The  equation  of  the  tangent  at  the  vertex  is,  therefore, 

(2/ -.44)  =1(0; -.08) 
which  reduces  to  4  a;  _  3  ?/  +  1  =  0. 

The  student  should  construct  a  figure  to  illustrate  this  example. 

EXERCISES 

1.  Find  the  equations  of  the  axes  of  the  ellipse 

a;2  -  2  xi/  +  4  2/2  +  2  X  +  10  1/  +  10  =  0. 

2.  Find  the  equations  of  the  axes  and  the  equations  of  the  asymptotes  of 
the  hyperbola  a;^  _  7  xy  +  2/2  +  12  x  +  3  2/  +  171  =  0. 

3.  Find  the  equation  of  the  axis,  of  the  tangent  at  the  vertex,  and  of  the 
directrix  of  the  parabola 

x2  -  2  X2/  +  2/"^  -  10  X  -  6  2/  +  25  =  0. 

4.  In  the  general  equation  of  a  conic,  show  that  the  line  gx  +  fy  +  c  =  0 
is  the  polar  line  of  the  origin  with  respect  to  the  conic. 

*  That  the  tangent  at  the  vertex  is  the  only  tangent  perpendicular  to  the  diameter 
through  its  point  of  contact  follows  from  Art.  96.  We  there  saw  that  the  coordi- 
nates of  the  point  of  contact  are  -^  and  —  ,  where  hi  is  the  slope  of  the  tangent. 
Hence,  as  hi  increases  indefinitely,  the  point  of  contact  approaches  the  origin. 


Arts.  120,  121]       THE   SYSTEM   OF   CIRCLES  163 

SYSTEMS  OF   CONICS 

120.  The  pencil  of  conies.  If  U  and  V  denote  expressions  of  the 
second  degree  in  x  and  y  and  Jc  is  any  constant,  then  U-\-kV=0  is 
the  equation  of  a  conic  passing  through  the  p)oints  common  to  U=  0 
and  V=0. 

For  U'-{-kV=  0  is  of  second  degree  in  x  and  y  and  is,  therefore, 
the  equation  of  a  conic.  This  conic  passes  through  the  points 
common  to  U=  0  and  V=  0,  since  its  equation  is  satisfied  by  the 
coordinates  of  these  points.  When  k  is  allowed  to  vary,  we  obtain 
a  series  of  conies,  each  passing  through  the  common  points. 
This  series,  or  system,  of  conies  is  called  a  pencil  of  conies. 

The  parameter  k  can  be  chosen  so  that  the  conic  U+kV=0 
shall  satisfy  some  additional  condition,  for  example,  that  it  shall 
pass  through  a  given  point  in  the  plane. 

121.  The  system  of  circles  with  a  common  radical  axis.  Suppose 
that  U=  0  and  V=  0  are  the  equations  of  two  circles  ;  that  is, 

U=  x^  +  if  +  Ax  +  By  +  C=  0, 

(1) 

and  F=  x2  +  7/2  +  yli.i- -f  5i?/ -f- Ci  =  0, 

then 

(x'  +  f  +  Ax  +  By  +C)A-  k{x-  +  f  +  A-->^'  +  B,y  +  C,)=0      (2) 

is  in  general  the  equation  of  a  circle  passing  through  the  common 
points  of  the  two  given  circles.  But  if  A:  =  —  1,  the  terms  of  sec- 
ond degree  drop  out,  and  (2)  becomes 

(A-A,)x+{B-B,)y  +  (C-C,)  =  0,  (3) 

which  is  of  first  degree  in  x  and  y,  and  therefore  the  equation  of  a 
straight  line.  This  line  is  called  the  radical  axis  of  the  system  of 
circles  U+kV=0.  The  radical  axis  is  a  real  line,  whether  the 
circles  intersect  in  real  points  or  not.  If  the  circles  intersect  in 
real  points,  the  radical  axis  is  the  common  chord. 

EXERCISES 

1.  Find  the  equation  of  the  conic  whicli  passes  through  the  points  common 
to  the  conies  a;^  —  Sxy  +  y^  —  Qx  =  0  and  4  x^  —  y'^  +  3  —  0,  and  also  through 
the  point  (3,  —  2) . 


164      EQUATIONS   NOT   IN   STANDARD   FORM    [Chap.  VIII. 

2.  Find  the  equation  of  the  radical  axis  of  each  of  the  following  pairs  of 
circles  : 

(a)   (X  -  2)2  +  (2/  -  3)2  -  10  =  0,   (x  +  3)2  +  {y  +  2)2  -6  =  0. 
(6)  x2  +  2/2  _  4  y  =  0,   (X  -  3)2  +  ?/  -  9  =  0 . 

(c)  (x  +  3)2  +  2/2  _  2  2/  _  8  =  0,  x2  +  2/2  -  2  ?/  =  0. 

(d)  x2  +  (2/  -  a)2  =  c2,  (x  -  2)2  +  2/^  =  cV. 

3.  Three  circles,  taken  in  pairs,  have  three  radical  axes.  Show  that  these 
radical  axes  intersect  in  one  and  the  same  point.  This  point  is  called  the 
radical  center  of  the  three  circles. 

4.  Find  the  coordinates  of  the  radical  center  of  the  three  circles 
(oc  _  3)2  ^  2/2  =  16,  x2  +  2/2  -  9,  and  x2  +  («/  -  2)2  =  25.  Construct  the  figure 
illustrating  this  exercise. 

5.  Show  that  the  length  of  a  tangent  from  the  point  (xi,  y{}  to  the  point 
of  contact  on  the  circle  x2  +  2/2  +  Z>x  +  ^(/  +  -^  =  0  is 


Vxi2  +  2/i2  +  Dxi  +Eiji  +  F. 

Suggestion.  The  triangle  whose  vertices  are  the  center  of  the  circle,  the 
point  of  contact,  and  the  point  (xi,  2/1)  is  right-angled  at  the  point  of 
contact. 

6.  Prove  that  the  locus  of  a  point,  the  lengths  of  tangents  from  which 
to  two  fixed  circles  are  equal,  is  the  radical  axis  of  the  two  circles. 

7.  Show  that  the  radical  axis  of  two  circles  is  perpendicular  to  the  line 
joining  the  centers  of  the  circlee. 

122.  The  parabolas  in  the  pencil  U+JcV=0.  If  the  constant 
]c  is  chosen  so  that  the  terms  of  second  degree  in  U'+JcV=0  form 
a  perfect  square,  the  corresponding  conic  is  in  general  a  parabola. 
But  it  may  be  a  pair  of  parallel  lines,  a  pair  of  imaginary  lines, 
or  a  single  line  (Arts.  115  and  116).  Since  the  condition  for  the 
parabola  is  of  second  degree  in  the  coefficients  of  x^,  xy,  and  /, 
there  are  in  general  two  parabolas  in  every  pencil  of  conies. 

For  example,  consider  the  pencil  of  conies  determined  by  the 

circle  U=x' +  y"" -l^x-d>y +  U  =  0, 

and  the  hyperbola     7-==  ^^2  _  ^2_  §0;  + 12  =  0. 
Here  the  equation 

(aj2  _|.  ^2  _  16  a;  _  8  2/  +  44)  +  A:(a;2  _  ^/^  -  8  .t  +  12)  =  0 

can  be  the  equation  of  a  parabola  only  if   k  =  ±  1.      Therefore 
the  pencil  contains  the  two  parabolas 

aj2_  i2a;-4?/  +  28=0  and  ?/2-4a;-4?/  +  16  =  0. 


Arts.  122,  123]     STRAIGHT   LINES   IN    U  +  kV  =  0 


165 


Fig.  93  illustrates  tlie  ex- 
ample. The  circle  and  hy- 
perbola have  but  two  real 
points  in  common  and  the 
two  x^arabolas  pass  through 
these  points. 

EXERCISES 

1.  Find  the  equations  of  the 
parabolas  which  pass  through 
the  points  common  to  the  circle 
x'^  +  y^  —  x— 9  =  0  and  the  hy- 
perbola oi'y  —1  =  0. 

2.  Find  the  equations  of  the 
two  parabolas  which  pass 
through   the   points   where   the 

ellipse  x!^  —  Sxy  +  iy^  —  X  —  2  =0  cuts  the  coordinate  axes.  (The  equation 
of  the  coordinate  axes  is  xy  =  0.)  Construct  the  figure  illustrating  this  ex- 
ercise. 

123.  Straight  lines  in  the  pencil  JJ+  kV=0.  AVhen  k  is  chosen 
so  that  the  discriminant  of  U+lcV=0  vanishes,  the  correspond- 
ing conic  is  in  general  a  pair  of  lines  (Art.  114). 

For  example,  consider  the  pencil  of  conies  determined  by  the  ellipses 
l/=  2x^  +  xy  +  6  2/2  +  3;  -  6  =  0  and  V=  3x^  +  6xy  +  10  y-2  +  x  -  10  =  0. 
Here  the  pencil  U+  kV  =  0  is 

(2  -h  3  k)x^  H-  (1  +  5  k)xy  -|-  (6  -h  10  k)y^  +(i  +  k')x-(6  +  10  k)  =  0. 

Forming  the  discriminant, 

1  +  5k 


FiCx.  93 


A  = 


(2  -h  3  k) 

l  +  5k 

2 

1  +  k 


2 
+  10  k) 

0 


1  +k 
2 

0 


(6  +  10  k) 


we  find  that  it  reduces  to 


-  12(6+ 10  k){k  +  l){2  k  +  I). 

Hence,  if  k  is  —  -|,  —  1,  or  —  J,  the  discriminant  is  zero  and  the  correspond- 
ing conic  consists  of  a  pair  of  lines.     If  A  =  —  |,  U  +  kV  =  0  becomes 

x^—lOxy  +  2?;  =  0, 


166      EQUATIONS  NOT   IN  STANDARD   FORM    [Chap.  VIII. 


which  is  the  equation  of  the  pair  of  lines 

X  =  0  and  x  —  lOy  +  2  =0. 

These  are  the  lines  BC  and  AD  in  Fig.  94.     It  k  = —1,   U+ kV=0  be- 
comes a;2  +  4  3:;/  +  4  2/2  -  4  =  0, 

which  is  the  equation  of  the  pair  of  parallel  lines 

x+  2y  -2  =  0  and  x  +  2y  +  2  =  0, 


1  1 

/V' 

^^ 

!S, 

,^: _  

: =■;             - 

><?         ^'^ 

i           ^ 

7     . 

:                          ^s^  Ji^' 

,'' 

^.                              ^s^ 

-^                           ^^ 

^^ 

^..    .=__._i. 

^,?_                           + .. 

:j.   .^,m_i^ 

I^i:    :::::::::::::::::::;::: 

^       >,i^-                                    :> 

--  =  '"T'Ti^             0        ^ 

^'"^             :^,       :  / 

•n 

-■^ 

?                                                           '*•«.,//'' 

^^^. 

"-'             -                              2V^ 

,^...                                                       ^t              s^ 

7 

■*;  j^ 

7 

"^-.^ 

-7 

"^^ 

:    '=^. 

Fig.  94 

or  BD  and  ^Cin  the  figure.     If  ^•  =—  i,  Z7+  A;F=  0  becomes 

a;2  _  3  x?/  +  2  2/2  +  X  —  2  =  0, 
or  (.r  -  2?/ +  2)(x— ?/ -  1)=  0, 

which  is  the  equation  of  the  pair  of  lines  AB  and  01). 

The  coordinates  of  the  points  A,  B.,  C,  and  B  are  now  easily  found, 
represent  the  common  solutions  of  the  equations  17=  0  and  Y=  0. 


They 


EXERCISES 

1.  Find  the  equations  of  the  straight  lines  which  join  in  pairs  the  points 
common  to  the  following  pair  of  conies  : 

(ff)  X?  +  2/'-  -25  =  0,  5  x2  +  14  2/  +  3  x  -  110  =  0. 

(6)  4  x2  +  9  2/  -  36  =  0,  x2  +  4  2/  =  0. 

(c)  x2  +  2  X2/  +  7  2/2  -  24  =  0,  2x^  -xy-f--^=  0, 

2.  Find  the  coordinates  of  the  points  common  to  each  pair  of  conies  in 
exercise  1. 


Art.  124] 


PENCIL   OF   CONICS 


167 


124.   The  pencil  of  conies  through  four  given  points.     In  the  pre- 
ceding article  we  have   seen  how  tlie  coordinates  of  the  points 
common  to  two  conies  U=  0  and  V^^  0  may  be  found.     On  the 
other  hand,  if  we  are  given  the 
coordinates  of  four  points,  we  can 
determine   tlie    pencil  of   conies 
which  has  these  points  in  com- 
mon. 

For    example,    let   the   four   given 

points  be  ^(1,  0),  5(2,  1),   C(l,  2), 

andZ>(0,  1)  (Fig.  95).     The  equation 

of  the  pair  of   lines  AB  and   CD  is 

then 

{x-xj-\){x-y  +  1)=  0, 

and  the  equation  of  the  pair  ^IC  and 

Therefore  the  equation  of  the  pencil  of  conies  having  the  four  given  points  in 

common  is         ,  ,,,  ,   -,\  ,    i  ,        is ,         i\      n. 

(x-y  -l)(x-y  +  l)+k(x-  l)(y-  1)=0. 

Clearly,  the  parameter  k  can  be  determined  so  that  the  corresponding  conic 
shall  pass  through  any  fifth  point  in  the  plane.  Thus,  if  we  wish  the  conic 
of  the  pencil  wliich  passes  througli  tlie  origin,  we  must  determine  k  so  that 
the  above  equation  shall  be  satisfied  by  the  coordinates  of  the  origin.  But 
the  equation  is  satisfied  for  x  =  0  and  y  —  0  H  kis  1.     Therefore  the  ellipse 

x^  —  xy  +  y'^  —  X  —  y  =  0 
belongs  to  the  pencil  and  also  passes  through  the  origin. 


EXERCISES 

1.   Find  the  equations  of  the  conies  which  pass  through  the  following  sets 


of  points 


(a)  (0,  0),  (1,  0),  (2,  1),  (1,  3),  and  (-  1,  -4). 

(b)  (1,  1),  (3,2),  (0,4),  (-4,0),  and  (-2,  -2). 


MISCELLANEOUS    EXERCISES 

1.  Show  that  the  line  x  —  2y  =  0  touches  the  circle 

x^  +  y^  —  4:X  +  8y  =  0. 

2.  The  line  ?/  =  3  x  —  9  is  tangent  to  the  circle 

x"^  +  y^  +  2  X  +  4:  y  —  5  =  0. 
Find  the  coordinates  of  tlie  point  of  contact. 


168      EQUATIONS  NOT   IN   STANDARD   FORM    [Chap.  VIIT. 

3.  Prove  that  the  distances  of  two  points  from  the  center  of  a  circle  are 
proportional  to  the  distances  of  each  from  the  polar  line  of  the  other. 

4.  Find  the  equations  of  the  circles  which  pass  through  the  intersections  of 

x^  +  2/2  =  9  and  x"^  +  y-  +  x+  2y  —  li 
and  touch  the  X-axis. 

5.  Find  the  coordinates  of  two  points  whose  polar  lines  with  respect  to  the 
circles  x'^  +  y^  _  2 x  —  3  =  0  and  x'^  +  y'^  +  2x—  17  =0  coincide. 

6.  Find  the  coordinates  of  the  radical  center  of  the  three  circles 

x2  +  2/2  _  4  a;  _  8  2/  -  5  =  0,  x2  +  z/2  -  8  X  -  10  2/  +  25  =  0, 
and  ft;2  +  2/2  +  8  X  +  11 2/  -  10  =  0. 

7.  Reduce  the  following  equations  to  a  standard  form  : 

(a)   (4  2/-3x)2  +  4(4x+32/)  =  0. 

(5)  4x2-24x2/+ 112/2- 16x- 2  2/- 89  =0. 

(o)  5  x2  -  4  X2/  +  8  2/2  -  22  X  +  16  2/  -  10  =  0. 

(d)  9  x2  -  12  X2/  +  4  2/2  =  10(2  x  +  3  2/  +  5). 

(e)  3  x2  -  2  X2/  +  2  2/2  -  16  X  -  8  2/  +  8  =  0. 
(/)  6  x2  +  24  x?/  -  2/^  +  50  2/  -  55  =  0. 

(g)  x2  —  2  X2/  +  2/2  —  5  X  —  2/  —  2  =  0. 

(/i)4x2  +  4x2/  +  2/^  +  4x  —  32/-|-4  =  0. 

(i)    25  x2  -  20  X2/  +  4  2/2  +  5  X  -  2  2/  -  6  =  0. 

( j)   x2  -  6  x?/  +  9  2/2  -  2  X  +  6  2/  +  1  =  0. 

(yfc)  x2  —  2xy-y--  20. 

{I)    xy  +  3  X  -  5  2/  +  5  =  0. 

(m)  x2  +  2  X2/  +  2/2  +  1  =  0. 

(n)   (5?/  +  12x)2  =  102x. 

(o)   x2  —  x?/  —  2  2/2  —  X  —  4  2/  —  2  =  0. 

8.  Wliat  curve  must  be  used  as  a  pattern  for  cutting  elbo-ws  of  stovepipes 
from  sheet  iron  ? 


CHAPTER   IX 
LOCI  OF  HIGHER  ORDER  AND  OTHER  LOCI 

125.  Certain  loci  of  higher  order,  as  well  as  certain  transcen- 
dental loci,  are  of  importance  either  because  they  are  useful  in 
mechanics  or  because  of  their  historical  interest.  The  more  im- 
portant of  these  loci  are  considered  in  the  following  articles. 

ALGEBRAIC   LOCI 

126.  The  Cissoid  of  Diodes.  Let  G  be  the  center  of  a  circle  of 
radius  a,  and  OCA,  any  diameter  of  it.  Through  0  draw  any 
chord  OR  and  produce  it  until  it  meets  the  tangent  at  A  in  the 
point  Q.  If  P  is  so  chosen  that  PQ  is  equal  to  OR,  then  the 
locus  of  P  is  a  curve  called  the  Cissoid  of  Diodes. 

To  find  the  equation  of  the  cissoid,  let  0  be  the  origin,  OCA 
the  X-axis,  and  the  tangent  at  0  the  Y-axis.  Let  6  denote  the 
angle  AOQ.     Then  OQ  =  2  a  sec  ^  and  OR  =  2  a  cos  6.     Hence, 

OP=OQ-  PQ^  Oq-  0R  =  2a (sec  d -  cos  6).  (1) 

But  OP  =  Vic^  -f-  ''f  and  d  =  arc  tan  ^  • 


Therefore  sec  0  = ^^-=^, 

X 

and  cos^  =  —  • 

Substituting  in  (1)  and  reducing,  we  get  the  equation  sought, 

y^  =  7T-^ — 

2  a  — X 

Either  from  the  definition,  or  the  equation,  the  curve  is  seen 
to  have  the  form  indicated  in  the  figure. 

169 


170 


LOCI   OF  HIGHER   ORDER 


[Chap.  IX. 


EXERCISES 

1.  Show  that  the  line  2  «  —  x  =  0  is  an  asymptote  to  the  cissoid. 

2.  Using  the  method  of  Art.  118,  show  that  the  tangent  to  the  cissoid  at 
the  point  (xi,  yi)  is  2(2  a  —  Xi)ijiy  —  (3  Xi^  +  yi^)x  +  2  ay{^  =  0. 

3.  In  Fig.  96  let  C3I  be  taken  twice  the  length  of  CB.  Draw  3IA  and 
let  it  meet  the  cissoid  in  the  point  F  whose  ordinate  is  FE.     Prove  that 

FF^  =  2  ■  0E\     If  CM  is  n  times  CB,  show 
that  FE^  =  n-  OE^. 

Note.  The  cissoid  was  invented  by  Dio- 
des for  the  purpose  of  duplicating  the  cube. 
Thus,  in  Fig.  96,  Avhen  CM  is  twice  CB, 
and  OF  is  the  edge  of  a  given  cube,  FE  is 
the  edge  of  a  cube  of  twice  the  volume. 
The  duplication  of  the  cube  is  one  of  the 
famous  problems  of  antiquity.  Diodes  lived 
about  150  B.C. 

127.    The   Conchoid   of  Nicomedes. 

Let  XX'  be  any  straight  line  and  0 

any    point   not    on    XX'.     Throngh. 

0   draw   a    series    of    straight   lines 

forming   a   pencil,    and   on    each  of 

these  lines  lay   off   a  constant  length  a  on  each  side  of  XX'. 

The  locus  of  the  points  so  determined  is  called  the  conchoid  of 

Nicomedes. 

To  find  the  equation  of  the  conchoid,  let  XX  be  the  X-axis  and 
the  perpendicular  through  0,  the  y-axis.  The  point  of  intersec- 
tion A  is  the  origin.  Let  OA  =  6,  and  P,  any  point  on  the  con- 
choid. Construct  the  right  triangle  POD,  PO  and  PD  meeting 
XX'  in  i^and  £",  respectively.     From  similar  triangles,  we  have 

EP:FE::DP:  OD. 


Fig.  96 


Now,  EP=y,  DP  =  11^  h,  and  OD  =  x. 
and  hence  FE 


Va^  —  y' 
also  for  the  point  P',  where  P'F=FP=a 
and  reducing,  we  have  the  equation  sought ;  namely, 


By  construction,  PF=a, 

It  is  clear  that  these  relations  hold 

Substituting  in  (1) 


x^y^  =  (y  -\-  h)-{a?  —  y"^). 


Arts.  127,  128]  THE   WITCH   OF   AGNESI 


171 


EXERCISES 

1.  Construct  conchoids  for  which  a  >  6,  a  =  &  and  a  <  6.  Note  the  differ- 
ence in  form. 

2.  Show  that  the  X-axis  is  an  asymptote  of  tlie  conchoid. 

3.  In  Fig.  97,  let  AB  be  twice  the  length  of  OF.  Draw  the  perpendicu- 
lar XX'  at  F  and  let  it  meet  the  conchoid  at  K.  Draw  OK  and  take 
KB  =  OF.      Show    that 

KB  =  RF  =  OF,  and  con- 
sequently the  angle  KOB 
is  one  third  the  angle 
FOB.  Show  how  this 
construction  enables  one 
to  trisect  any  given  angle. 
Note.  The  conchoid 
was  invented  by  Nieome- 
des  for  the  purj)ose  of  tri- 
secting a  given  angle. 
This   is   another   famous  Fio.  07 

problem     of     antiquity. 

Neither  the  duplication  of  the  cube  nor  the  trisection  of  an  angle  can  be 
effected  by  means  of  the  circle  and  straight  line  alone,  hence  the  ancients 
were  forced  to  the  invention  of  other  curves  for  these  purposes.  Nicomedes 
was  a  contemporary  of  Dio.cles. 

128.    The  Witch  of  Agnesi.     Let  G  be  the  center  of  a  circle 
whose  radius  is  a ;  and  OB  any  diameter.     Draw  the  tangents  at 
^-  0  and  B ;  and  let 


OR  be  any  chord 
through  0  which, 
produced,  meets 
the  tangent  at  B  in 
N.  Through  R 
draw  the  parallel 
to  OD,  the  X-axis, 
and  through  N,  the  parallel  to  OB,  the  Y-axis.  The  locus  of 
the  point  P,  where  these  parallels  meet,  is  called  the  witch  of 
Agnesi. 


Fig.  98 


172 


LOCI   OF  HIGHER   ORDER 


[Chap.  IX. 


EXERCISES 

1.    Show  that  the  equation  of  the  witch,  referred  to  the  lines  OX  and  OY 
8a3 


as  coordinate  axes,  is  y 


+  4a2 


Suggestion.  Use  the  similar  triangles  NBP  and  NOD  to  find  the  re- 
lation between  the  coordinates  of  P. 

2.    Shpw  that  the  X-axis  is  an  asymptote  of  the  witch. 

Note.  Donna  Maria  Agnesi,  who  invented  the  witch,  was  horn  at  Milan, 
1718,  and  died  there,  1799.  She  was  appointed  Professor  of  Mathematics 
at  the  University  of  Bologna,  1750. 

129.  The  Limacon  of  Pascal.  Let  C  be  the  center  of  a  circle 
whose  radius  is  a ;  and  OD,  any  diameter.  Through  0  draw  a 
series  of  lines,  and  on  each  of  these  lay  off  a  distance  b  on  each 
side  of  the  circle.  The  locus  of  the  points  thus  determined  is 
called  the  limacon. 

To  find  the  equation  of  the  limaqon,  let  0  be  the  pole  and  OD 
the  polar  axis.  The  length  of  the  chord  within  the  circle  is 
2  a  cos  6.  Hence  the  radii  of  the  points  P  and  P'  on  this  chord 
are  given  by  the  equation 

r  =  2  a  cos  0  ±  b, 

which  in  rectangular  coordinates  reduces  to 

(a;2  +  /  -  2  axf  =  b'  (x'  +  /). 


EXERCISES 

1.  Construct  the  limagons  for 
which  6  >  2  a,  &  =  2  a,  and  6  <  2  a. 
Note  the  difference  in  form.  When 
&  =2  a,  the  limacon  is  called  the  car- 
dioid  from  its  heart-shaped  form. 

2.  When  b  =  a,  the  limagon  fur- 
nishes a  neat  curve  for  trisecting  a 
given  angle.  In  Fig.  99,  let  PCB  be 
the  given  angle.  Show  that  PM  = 
MC  =  CO  and,  therefore,  the  angle 
POB  is  I  the  angle  PCB. 

Note.  Pascal  (1623-1662)  was  a 
celebrated  French  mathematician 
and  philosopher. 


Fig.  99 


Arts.  129,  130]  THE   CYCLOID  173 

MISCELLANEOUS   EXERCISES 

1.  Show  that  the  locus  of  the  intersection  of  a  tangent  to  the  parabola 
2/2  =—  8  ax  and  a  line  drawn  through  the  origin  perpendicular  to  this  tangent 
is  the  cissoid. 

SuGGESTiox.     The  equation  of  a  tangent  in  terms  of  the  slope  is  (Art.  95, 

2  a 
Eq.  9)  2/  =  inx  — ,  and  the  equation  of  the  line  through  the  origin  per- 
pendicular to  this  tangent  is,  y  — '-■     The  locus  of  the  intersection  of  these 

m 

two  lines  is  found  by  eliminating  m  from  the  two  equations. 

2.  A  tangent  is  drawn  to  the  parabola  2/^  =  4  px  at  a  point  T.  The  per- 
pendicular to  this  tangent  through  the  origin  meets  the  ordinate  of  T,  pro- 
duced, at  P.  Find  the  equation  of  the  locus  of  P  as  T  moves  along  the 
curve.     The  locus  is  called  the  semicubical  parabola. 

3.  The  two  parabolas  j/^  =  2  ax  and  x^  =  ay  meet  at  the  origin  and  also 
at  another  point  P.  Find  the  coordinates  of  P.  If  a  is  the  edge  of  a  given 
cube,  show  how  the  construction  of  the  two  parabolas  solves  the  problem  of 
the  duphcation  of  the  cube. 

4.  Show  that  the  conchoid  is  the  locus  of  the  points  of  intersection  of  the 
line   y  =- —  b    with    the    circle    (x  —  bky'+ y'^  =  a-,    k    being    a    variable 

^■ 

parameter. 

5.  A  tangent  is  drawn  to  the  equilateral  hyperbola  ^"  —  y'^  —  cfi  at  the 
point  T.  The  perpendicular  to  this  tangent  through  the  origin  meets  the 
tangent  in  the  point  P.  Show  that  the  locus  of  P,  as  T  moves  along  the 
curve,  is  the  lemniscate  (Art.  54). 

6.  Find  the  locus  of  the  intersection  of  the  two  straight  lines 

X  -\-  ky  +  a{k'^  —  3)=  0  and  y  =  kx, 

k  being  a  variable  parameter.    The  locus  is  called  the  trisectrix  of  Maclaurin. 
Discuss  its  equation  and  draw  the  locus. 

TRANSCENDENTAL   LOCI 

130.  The  cycloid.  The  locus  of  a  point  in  the  circumference 
of  a  circle  which  rolls  (without  sliding)  along  a  fixed  straight  line 
is  called  the  cycloid.     The  circle  is  called  the  generating  circle. 

To  find  the  equation  of  the  cycloid,  take  the  line  on  which  the 
circle  rolls  as  the  X-axis,  the  radius  of  the  rolling  circle  equal  to 
a,  and  one  of  the  positions  in  which  the  tracing  point  is  on  this 
line  as  the  origin  0.  Let  C  be  the  center  of  the  generating  circle 
when  the  tracing  point  has  the  position  P.     Join  P  and  C,  and 


174 


LOCI   OF  HIGHER   ORDER 


[Chap.  IX. 


draw  CT  perpendicular  to  the  X-axis.     Let  9  represent  the  angle 

PGT.     Now  or  =  arc  Pr=a^.     Hence, 

x^OD=OT-PE  =  ad-a  sin  d, 

y  =  DP=  TC -  EC  =  a- a  cos  B.  ^^^ 

Equations  (1)  are  the  parametric  equations  of  the  cycloid,  0 
being  the  parameter. 

When  0  varies  from  0  to  2  tt,  P  traces  out  one  arch  of  the  ciirve. 
The  entire  curve  consists  of  this  arch  and  repetitions  of  it  to  the 
right  and  left  corresponding  to  values  of  9  outside  the  limits  0,  2  tt. 


EXERCISES 

1.  In  Fig.  100,  OB  is  tlie  span  of  one  arcii  of  the  cycloid  and  F  is  its 
middle  point.  Show  that  the  area  of  the  triangle  OAB  is  twice  the  area  of 
the  generating  circle. 

2.  When  the  point  T  bisects  OF,  show  that  the  area  of  the  triangle  OPA 
is  half  the  area  of  the  square  inscribed  in  the  generating  circle. 


Fig.  100 


3.  Prove  that  the  tangent  to  the  cycloid  at  P  passes  through  H,  the  upper 
extremity  of  the  diameter  of  the  generating  circle  which  is  perpendicular  to 
the  base  OB. 

Suggestion.  At  any  instant  of  the  motion  of  the  generating  circle,  T 
(its  lowest  point)  is  at  rest,  and  the  motion  of  every  point  of  the  generating 
circle  is  for  the  moment  the  same  as  if  it  described  a  circle  about  T.  Hence 
the  normal  to  the  cycloid  at  P  must  pass  through  T. 

4.  If  ^  =  -,  write  the  equation  of  the  tangent  to  the  cycloid. 

Note.  The  cycloid  was  much  studied  by  the  most  eminent  mathema- 
ticians of  Europe  during  the  first  half  of  the  17th  century.  In  particular 
Galileo  and  Pascal  discovered  many  of  its  properties.  Its  area  was  found  to 
be  three  times  the  area  of  the  generating  circle  by  Roberval  in  1634.  The 
method  of  drawing  tangents  (exercise  3)  was  discovered  by  Descartes. 


Art.  131] 


THE  HYPOCYCLOID 


175 


131.  The  Hypocycloid.  This  locus  is  the  curve  traced  by  a 
fixed  point  on  the  circumference  of  a  circle  which  rolls  internally 
along  the  circumference  of  a  fixed  circle. 

To  derive  the  parametric  equations  of  the  hypocycloid,  let  the 
radii  of  the  fixed  and  rolling  circles  be  a  and  b  respectively. 
Let  A  be  one  of  the 
positions   in    which  ^ 

the  tracing  point  lies 
on  the  fixed  circle. 
Take  the  center  of 
the  fixed  circle  0  as 
origin,  and  the  line 
OA  as  X-axis.  Let 
C  be  the  center  of 
the  rolling  circle 
when  the  tracing 
point  has  arrived  at 
the  position  P(;x,  y), 
6  the  angle  througli 
which    the    line    of  Fig.  101 

centers     OCB     has 

turned,  and  ^  the  angle  through  which  the  radius  CP  of  the  rolling 
circle  has  rotated  since  P  left  the  position  A.  The  coordinates  of  P 
are  the  coordinates  of  C  plus  the  projections  of  CP  upon  the  X- 
and  F-axes,  respectively.     Hence  (Fig.  101),  we  have 

X  =  0M=  0H+  NP=  OCcos  ^  +  CP  cos  <^  =  (a  -  6)cos  ^  +  &  cos  c^, 

y  =  MP  =  nC-  NC  =  (a  -b)s\nO-b  sin  <^. 
But  arc  PB  =  arc  AB,  and  therefore  we  have 

b{6  +  <l,)=ae,  or  ^  =  "-1^6. 

Hence,  the  required  equations  are 

a;  =  (a  —  b)  cos  d  +  b  cos 

y  =(ci  —  b)  sin  0  —  b  sin 


a 

— 

b 

6 

b 

a 

— 

b 

e 

b 

176 


LOCI   OF  HIGHER   ORDER 


[Chap.  IX. 


132.    Special  Hypocycloids.     If  a  =  2  &,  the  equations  of  the 
hypocycloid  become 

x  =  2'b  cos  B, 
y  =  0. 

Hence,  in  this  case,  the  hypocycloid  consists  of  that  portion  of 
the  X-axis  which  is  included  within  the  fixed  circle. 
If  a  =  4  6,  the  equations  become 


i»  =  i^(3cos^  +  cos3^), 
4 

y  =  -{3  sin  6  -  sin  3  6). 
4 

But,  from  trigonometry, 

cos  3^  =  4  cos^  ^  —  3  cos  6, 

sin  3  ^  =  3  sin  ^  -  4  sin^  0. 

Substituting  these  values  in  (1),  we  get 


(1) 


x  =  a  cos^  6, 

y  =  a  sin^  6, 

fro 

m  which 

cos  6  = 

($!■ 

sin  6  = 

©* 

Squaring,  adding,  and 
clearing    of    fractions. 

we 

have 

Fig.  102 


.1-3  +  2/'  =  a3. 

The  tracing  point  on 
the  rolling  circle  re- 
verses its  direction  of 
motion  at  each  of  the  positions  in  which  it  is  in  contact  with  the 
fixed  circle.  These  points  are  called  cusps.  Thus,  in  Fig.  101, 
the  points  A,  D,  E,  are  cusps. 

If  a  =  4  6,  there  are  four  cusps,  since  the  rolling  circle  makes 


Aets.  132,  133]  THE     EPICYCLOID  177 

exactly  four  complete  revolutions  in  returning  to  its  original  posi- 
tion.    Hence,  the  locus  is  called  the  four-cusped  hj^ocycloid.     It 

is  an  algebraic  curve,  since  its  equation,  x^  -j-?/^  =  a^,  is  algebraic. 
The  equation  is  readily  rationalized,  and  it  then  has  the  form 
(a;2  -|_  tf  _  a2)3  _  27  aV^-  =  0, 

from  which  it  is  seen  that  the  curve  is  of  6th  order.     The  form 
of  the  curve  appears  in  Fig.  102. 

EXERCISES 

1.  Show  that  the  tangent  and  normal  at  any  point  P  of  an  hypocycloid 
pass  through  the  extremities  of  that  diameter  of  the  rolling  circle  which 
passes  through  the  center  of  the  fixed  circle  (in  Fig.  101,  the  tangent  and 
normal  at  F  pass  through  F  and  B  respectively). 

2.  Prove  that  the  length  of  the  tangent  at  any  point  of  the  four-cusped 
hypocycloid,  which  is  included  between  the  coordinate  axes,  is  equal  to  the 
radius  of  the  fixed  circle. 

Let  F  (Fig.  102)  be  any  point  of  the  curve,  and  C,  the  center  of  the  roll- 
ing circle  when  the  tracing  point  has  the  position  F.  The  tangent  at  F  is 
then  FF,  meeting  the  axes  in  iiTand  L.  We  are  to  show  that  KL  is  equal 
to  the  radius  of  the  fixed  circle.  Since  arc  AB  =  arc  FB  and  the  radius  of 
the  fixed  circle  is  four  times  the  radius  of  the  rolling  circle,  it  follows  that 
the  angle  BCF  is  four  times  the  angle  BOA;  that  is,  angle  BCF  =  4  6. 
Hence,  angle  BFF  =  2d;  and  OFL  is  an  isosceles  triangle,  OF—FL. 
Also  OFK'iB  an  isosceles  triangle  and  OF  — -  FK.   Therefore  LK  =  2  •  OF— a. 

Note.  When  a  straight  line,  or  curve,  moves  according  to  a  given  law, 
it  is  generally  continuously  tangent  to  another  curve  called  the  envelope. 
Thus,  the  four-cusped  hypocycloid  is  the  envelope  of  a  line  of  constant 
length  which  moves  so  that  its  extremities  are  always  on  the  coordinate 
axes.  This  property  enables  one  to  construct  the  four-cusped  hypocycloid  by 
merely  drawing  a  series  of  lines  of  constant  length  whose  extremities  all  Ue 
on  the  coordinate  axes.     The  student  should  make  the  construction. 

133.  The  Epicycloid.  The  locus  is  the  curve  traced  by  a  fixed 
point  on  a  circle  which  rolls  externally  on  the  circumference  of  a 
fixed  circle. 

The  parametric  equations  of  the  epicycloid  are  found  in  exactly 
the  same  way  as  were  the  equations  of  the  hypocycloid  (Art.  131). 
They  may  be  written  from  the  equations  of  the  hypocycloid  by 
changing  the  sign  of  h.     The  equations  are 


178 


LOCI   OF   HIGHER   ORDER 


[Chap.  IX. 


x=  (a  +  b)  cos  6—  b  cos 
2/  =  (a  +  b)  sin  6  —  b  sin 


fa +6   1 

L     ^       J 

L    ^       J 

Fig.  103 

134.  The  Cardioid.  When  the  rolling  circle  is  equal  to  the 
fixed  circle ;  that  is,  when  a  =  b,  the  equations  of  the  epicycloid 

become 

X  =  2  a  cos  0—  a  cos  2  6, 
y  =  2a  sin  0  —  a  sin  2  0. 

To  show  that  this  curve  is  the  cardioid  (Art.  129,  exercise  1), 
let  Cbe  the  center  of  the  rolling  circle  when  the  tracing  point  has  the 
position  P.  Draw  PA  and  let  it  meet  the  fixed  circle  again  at  Q. 
Now,  since  arc  PF  =a,vG  AF  and  FC  =  FO,  the  angle  FCP  = 
angle  FOA  =  6 ;  FP  =  FA  and  PQ  is  parallel  to  OC.  Again, 
since  OA  =  OQ  and  angle  OAQ  =  0,  angle  OQA  =  0  and  QOCP 
is  a  parallelogram.  Therefore,  QP  =  OC  =  2  a,  and  the  point  P 
can  be  located  by  laying  off  the  distance  2  a  from  Q  along  the 
chord  of  the  fixed  circle  through  A.  This  agrees  with  the  defini- 
tion of  the  limaQon  for  which  b=  2  a.  Hence,  the  sjjecial  epicy- 
cloid for  ivhich  the  rolling  circle  is  equal  to  the  fixed  circle  is  the 


Art.  134] 


THE   CARDIOID 


179 


same  as  the  special  limacon  for  ivJiich  the  distance  laid  off  along  the 
chords  of  the  fixed  circle  is  twice  the  radius  of  the  fixed  circle. 


Fig. 104 

The  tangent  to  the  carclioid  at  P  passes  throngh  T,  and  the 
normal  through  F.  Hence,  the  circle  whose  center  is  F  and 
whose  radius  is  FP  touches  the  cardioid  at  P.  But  this  circle 
passes  through  A,  since  FP=:FA.  Therefore,  all  the  circles  hav- 
ing their  centers  on  the  fixed  circle  and  passing  through  A,  touch  the 
cardioid.  Or,  in  other  words,  the  cardioid  is  the  envelope  of  this 
system  of  circles.  This  property  enables  one  to  construct  the 
cardioid  by  drawing  a  number  of  circles  of  the  system. 

While  the  epicycloids  are  in  general  transcendental  curves,  the 
cardioid,  as  we  have  seen,  is  an  algebraic  curve. 

MISCELLANEOUS    EXERCISES 

1.    A  circle  of  radius  a  rolls  along  a  fixed  straight  line  ;  a  point  on  a 
fixed  radius  of  the  circle  at  a  distance  b  from  the  center  describes  a  curve 


180 


LOCI   OF  HIGHER   ORDER 


[Chap.  IX. 


called  the  trochoid.     Show  that  the  parametric  equations  of  the  trochoid  are 

X  =  ad  —  b  sin  0, 
y  —  a  —  b  cos  6. 

Plot  the  trochoids  for  which  b  <^a  and  b^a. 

2.  Show  that  the  polar  equation  of  the  cardioid  (Fig.  104)  is 

r  =  2  a{\  —  cos  6), 
A  being  the  pole  and  OA  the  polar  axis. 

3.  Write  the  parametric  equations  of  the  hypocycloid  for  which  a  —  3b. 
This  curve  is  called  the  three-cusped  hypocycloid. 

4.  A  thread  is  wound  around  a  circular  disk  and  then  unwound,  kept 
always  stretched.     Any  point  in  the  thread  describes  a  curve  called  the 

involute  of  the  circle.     If  a  is  the  ra- 
^T  dius  of  the  circle,  A  the  position  where 

the  tracing  point  leaves  the  circle,  O 
(the  center  of  the  circle)  the  origin, 
rp  I  and   OA  the  X-axis,  show  that  the 

parametric  equations  of  the  locus  are 

X  =  a  cos  6  +  ad  sin  6, 
y  —  a  sin  e  —  ad  cos  6. 

5.     A      ladder      stands     upright 
against  a  perpendicular  wall  and  then 
slides  down,  the  upper  end  continually 
resting  against  the   wall.      What  is 
Pjq  2Q5  the  envelope  of  the  moving  ladder  ? 

What  is  the  locus  of  its  middle  point  ? 

What  are  the  loci  of  the  points  dividing  the  ladder  in  the  ratio  —  ? 

n 

6.  A  projectile  leaves  the  muzzle  of  a  gun  with  a  velocity  of  v  feet  per 
second,  the  barrel  of  the  gun  being  elevated  at  an  angle  </>  from  the  horizontal. 
Neglecting  the  resistance  of  the  atmosphere,  show  that  the  path  of  the  pro- 
jectile is  a  parabola  whose  parametric  equations  are 


X  =  vt  cos  (p, 
y  —  vt  sin  <p  — 


2  ' 


the  parameter  t  denoting  time  measured  in  seconds,  and  g,  the  force  of  gravity. 

Take  the  position  of  the  gun  as  origin  and  the  horizontal  line  OL  as  X-axis. 

Now,  at  the  end  of  t  seconds,  the  projectile  would  be  at  Q{  OQ  —  vt)  were  it 

2 


not  for  gravity  which  pulls  it  down  a  distance  QP 


Hence, 


Art.  134] 


THE   CARDIOID 


181 


X  —  OB  =  vt  cos  <t> 
y  =  DP  =  DQ  -  PQ  =  vt  sin  < 
Eliminating  t  from  these  equations, 


2 


%l  =  X  tan  <b  —  4^,  cos-  (p 


Hence,  tlie  locus  is  a  parabola. 

7.  The  distance  OL  (Fig.  106)  from  the  gun  to  the  point  where  the  pro- 
jectile strikes  the  horizontal  line  is  called  the  range.  Derive  a  formula  for 
computing  the  range  when  the 
velocity  v  and  the  angle  of  ele- 
vation (p  are  given.  Show  that 
the  greatest  range  is  obtained 
wlien  0  =  45°.  If  the  velocity 
V  is  1000  ft.  per  second  and  the 
range  is  5  mUes,  what  must  be 
the  angle  of  elevation  ? 

8.  Two  men  compete  in 
putting  the  shot.  Compute  the 
effect  of  any  difference  in  height 
of  the  men,  other  things  being 
equal. 

9.  A  form  is  to  be  con- 
structed for  a  parabolic  arch  of  Fig.  106 
cement  work.     The  height  of 

the  arch  is  h  and  the  span  2  I ;   find  the  equation  of  the  arch. 

10.  Find  the  equation  of  the  locus  of  the  foot  of  the  perpendicular  from 
the  center  upon  the  tangent  to  the  ellipse 

11.  The  hypotenuse  of  a  right  triangle  is  given  in  position  and  length. 
Find  the  equation  of  the  locus  of  the  center  of  the  circle  inscribed  in  the 

triangle. 

12.  Find  the  locus  of  the  center  of  a  circle  which  touches  two  given 
circles.  Discuss  the  problem  for  the  various  positions  which  the  given  circles 
may  have. 

13.  Through  a  given  point  (xi,  yi)  two  lines  are  drawn  which  meet  the 
coordinate  ases  in  the  points  A,  B  and  Ai,  Bi,  respectively.  Find  the  locus 
of  the  point  of  intersection  of  the  lines  ABi  and  A^B. 


182  LOCI   OF  HIGHER   ORDER  [Chap.  IX. 


EMPIRICAL   EQUATIONS  AND   THEIR   LOCI 

Pairs  of  corresponding  values  of  two  variable  quantities  are 
often  found  by  experiment;  the  graph  or  locus  determined  by 
these  pairs  (Art.  40)  exhibits  the  change  in  function  due  to  a 
change  in  variable  within  the  limits  of  observation.  It  is  often 
of  importance  to  determine  the  equation  of  this  graph  or  locus; 
or,  to  speak  more  accurately,  to  find  an  equation  whose  locus 
coincides  as  closely  as  possible  with  the  locus  formed  from  the 
observed  pairs  of  values.  An  equation  found  in  this  way  is  called 
empirical,  because  it  depends  upon  experiment  or  observation. 
An  empirical  equation  is  the  mathematical  statement  of  an  em- 
pirical law. 

The  complete  solution  of  the  problem  of  finding  a,n  empirical 
equation  from  a  given  set  of  observations  often  leads  to  very 
intricate  analysis,  but  it  is  not  difiicult  to  test  a  set  of  observa- 
tions to  see  if  it  satisfies  any  one  of  a  number  of  typical  equations. 
These  typical  equations  embody  the  simple  laws  of  natural  science. 

135.  Typical  Equations. 

1.  y==  mx  +  k ;  straight  lines. 

2.  y=  Cx"\  parabolic  curves  (Fig.  107). 

3.  (y  —  Jc)  =  C (x  —  h)" ;  parabolic  curves,  origin  not  at  the 
vertex. 

4.  2/  =  —;  hyperbolic  curves  (Fig.  107). 

a?" 

5.  {jj  —  k)  = ;    hyperbolic    curves,    origin    not    at    the 

,  (  Jj  ft  1 

center,  ^  ^ 

,~^'      ,    i  ;  exponential  curves  (Fig.  109). 
(2/-A-)=a6^J  ' 

7.  2/=^^  (Fig.  110). 

8.  y  =  a  +  bx-\- cx^.  -\- dx? -!-•••+  kx^. 

136.  Loci  of  typical  equations.  A  study  of  the  foregoing  equa- 
tions and  the  general  characteristics  of  the  corresponding  loci  is 
of  fundamental  importance. 


Akts.  135,  136]     LOCI   OF  TYPICAL  EQUATIONS 


183 


Figure  107  shows  a  number  of  curves  belonging  to  the  family 
y  =  Cx\  This  is  an  especially  important  family  of  curves,  since 
many  of  the  simple  laws  of  natural  science  are  embodied  in  the 


Fig.  107 

equation  y  =  Cx".     The  curves  are  drawn  for  C=l;  the  general 
form  of  the  curves  will  be  the  same  for  any  other  value  of  C. 

If  n  is  a  positive  number  (type  2,  Art.  135),  the  curves  are 
called  parabolic.  If  n  is  3  or  i,  the  curve  is  the  cubical  parabola. 
If  rj  is  f  or  -|,  the  curve  is  the  semicubical  parabola.  If  n  is  2  or 
4-,  the  curve  is  the  ordinary  parabola.  If  n  is  a  negative  number, 
the  curves  are  called  hyperbolic.  If  n  is  —  1,  the  curve  is  the 
ordinary  hyperbola. 


184 


LOCI  OF  HIGHER  ORDER 


[Chap.  IX. 


A  study  of  the  figure  reveals  the  following  properties : 

I.  All  curves  of  the  system,  ivhatever  the  value  of  n,  2^ass  through 
the  point  A  =(1,  1). 

II.  All  the  parabolic  curves  of  the  system  (ii  >  0)  pass  through  the 
origin,  folloio  the  diagonals  of  the  square  ABCD  more  or  less  closely, 
and  pass  out  of  the  square  through  A  and  one  other  vertex. 

III.  If  n  is  a  positive  number  represented  by  -  (a  and  b  p)Time  to 
each  other),  then 

(1)  when  a,  is  even  and  b  is  odd,  the  curves  pass  through 

(2)  when  a  is  odd  and  b  is  odd,  the  curves  pass  through 

(7  =  (—  1,  —  1) ;  and 

(3)  wheti  a  is  odd  and  b  is  even,  the  curves  pass  through 

D^{1,-1). 

IV.  The  parabolic  curves  of  the  system  fill  the  square  ABCD  and 
the  infinite  regions  of  the  plane  which  corner  on  this  square  ;    i.e.  the 

shaded  regions  in  Fig.  108. 
V.  When  n  is  a  positive 
even  integer,  the  curves  touch 
the  X-axis  more  and  more 
closely  the  larger  n  is  taken  ; 
i.e.  the  curvature  at  the  ori- 
— ^      gin  becomes  less  and  less  as 

the  value  of  n  is  increased. 

When  ri-  is  a  positive  odd 
integer,  the  curves  touch 
the  X-axis  at  the  origin, 
but  the  curvature  changes 
from  concave  downward  on 
the  left  of  the  F-axis  to 
concave  upward  on  the  right.  Each  curve  has  a  point  of  inflexion 
at  the  origin. 

When  n  is  fractional,  with  neither  numerator  nor  denominator 
equal  to  unity,  each  curve  has  a  cusp  at  the  origin. 


B 


ir 


Fig.  108 


Art.  136.] 


LOCI  OF  TYPICAL   EQUATIONS 


185 


VI.  Tlie  hyperbolic  curves  of  the  system  (n  <  0)  Jill  the  regions  of 
the  plcme  outside  the  square  ABCD  and  the  infinite  regions  corner- 
ing on  this  square  ;  i.e.  the  un- 
shaded regions  in  Fig.  108.  The 
axes  are  asymptotes  to  each  hy- 
perbola of  the  system. 

Types  3  and  5  (Art.  135)  do 
not  differ  in  form  from  types  2 
and  4. 

Type  6  is  illustrated  in  Fig. 

109.  Each  curve  of  the  system 
passes  through  the  point  (0,  1) 
(a  is  assumed  to  be  unity  in 
drawing  these  curves),  the 
curvature  depending  upon  the 
value  of  b.  The  curves  illus- 
trate phenomena  that  follow  the 
"compound  interest  law." 

Type    7    is    shown    in    Fig. 

110.  If  a^  =  &',  the  curve  is  the  witch  (Fig.  98).  This  curve 
is  of  special  importance  in  representing  phenomena  where  the  ob- 
served value  (the  function)  gradually  decreases,  from  a  maximum 


Fig.  109 


y=bTT.  ;«=4,6=1 


Fig  110 


186 


LOCI   OF  HIGHER   ORDER 


[Chap.  IX. 


at  X  —  0,  as  the  variable .  increases  in  value.     Other  curves  ap- 
plicable to  phenomena  of  this  character  are  the  probability  curve, 


Fig. Ill 


A; 


y=  — =6"'^'"'^',  Fi^.  Ill,  and  the  curve  y  = 

a  =  1,  the  latter  is  the  hyperbolic  secant  curve. 


-,  Fig.  112.     If 


Fig.  112 

137.    Selection    of  type  curve   and   determination   of   constants. 

Frequently  the  law  which  experimental  data  must  follow  is 
known  beforehand,  and  then  it  is  only  necessary  to  determine 
the  constants  in  the  equation.  For  example,  in  experiments  on 
falling  bodies,  the  law  is  known  to  be  of  the  form  y  =  Cx^,  where 
y  represents  the  distance  fallen  during  the  time  x.     In  this  case, 


Arts.  137,  138]    TEST   BY   LINEAR  EQUATIONS  187 

one  pair  of  values  of  x  and  y  will  determine  the  value  of  the  con- 
stant C. 

Experimentally  determined  values  of  any  function  are  never 
absolutely  exact,  so  that  plotted  points,  determined  from  experi- 
ment, never  all  lie  exactly  on  the  curve  representing  the  known 
law.  The  values  of  the  constants,  determined  as  above,  are  there- 
fore more  or  less  approximate.  The  aim  is  to  find  such  values 
for  the  constants  as  will  give  the  best  average  curve  to  represent 
the  observed  values  of  the  function. 

In  case  the  law  is  not  known,  the  curve  which  best  represents 
the  observed  values  of  the  function  must  be  selected  by  trial. 
The  procedure  is  as  follows  : 

(a)  Plot  the  observed  values  carefully; 

(h)  From  the  known  forms  of  curves  discussed  in  Art.  136, 
or  elsewhere,  select  that  one  which  resembles  the  plotted  curve 
most  closely ; 

(c)  Determine  the  constants  in  the  equation  of  the  selected 
curve  so  that  it  will  fit  the  observed  values  most  closely. 

To  fulfill  the  requirements  (6)  and  (c)  satisfactorily  requires 
good  judgment  and  a  good  eye  as  well  as  some  knowledge  of  the 
forms  of  various  types  of  curves.  The  results  obtained  are  often 
quite  as  serviceable  as  though  more  intricate  analysis  had  been 
employed  to  find  them. 

138.  Test  by  means  of  linear  equations.  After  a  trial  curve  has 
been  selected,  it  is  often  rather  difficult  to  determine  whether 
this  curve  actually  represents  the  observed  values  of  the  function 
with  sufficient  accuracy  for  the  purposes  of  the  problem  or  not. 
The  following  device  is  of  great  assistance  in  determining  whether 
to  retain  or  reject  the  trial  curve.  The  typical  equations  in 
Art.  136  can  be  transformed  into  linear  equations  as  follows : 

2.  ?/  =  Cx"  can  be  written  log  y  —  log  C  -f  n  log  x. 

3.  (y  —  k)=C{x  —  hy  can  be  written  log  (2/ —  A;)  —  log  C + 
n  log  (x  —  h). 

Types  4  and  5  are  included  in  the  above. 

6.  y  =  ab^  can  be  written  log  y  =  log  a  -f-  x  log  b. 

Ct  1         &         'V^ 

7.  y  = ,  can  be  written  -  =  -  -f  — . 

b  +  x"'  y      a      a 


188 


LOCI   OF  HIGHER   ORDER 


[Chap.  IX. 


If  the  observed  values  satisfy  sufficiently  accurately  an  equa- 
tion of  the  form  y  =  Cx^,  say,  then  the  logarithms  of  the  observed 
values  must  satisfy  the  equation  log  y  =  log  C  -{-  n  log  x.  Hence, 
the  points  plotted  from  the  logarithms  of  the  observed  values 
must  lie  closely  upon  a  straight  line.     If  they  do  not  lie  upon  a 


3       4     5   6  7  891 

Fig.  113 


4     5    6  7  6  91 


straight  line,  or  nearly  so,  the  required  equation  is  not  of  the 
form  y  =  Cx". 

Similarly,  if  the  observed  values  are  to  satisfy  an  equation  of 
the  type  y  =  alf,  then  the  values  of  log  y  and  x  must  satisfy  the 
equation  log  y  =  log  a -\- x  log  h. 

Again,  if  the  observed  values  are  to  satisfy  an  equation  of  the 


Art.  139] 


EXAMPLES  AND   EXERCISES 


189 


type  7,  the  values  of  -  and  x^  must  satisfy  the  equation 

y 

y     «     o 

Instead  of  looking  up  the  logarithms  of  the  numbers  in  a 
given  table,  logarithmic  paper  may  be  used.  The  horizontal  and 
vertical  scales  on  this  paper  represent  the  logarithms  of  numbers. 
Figure  113  shows  a  sheet  of  this  paper  on  which  has  been  plotted 
the  table  in  Example  II,  Art.  139.  Because  the  plotted  points 
lie  very  closely  upon  a  straight  line,  we  may  assume  the  equa- 
tion y  =  Cx'^. 

139.  Examples  and  exercises.  The  foregoing  statements  will 
be  better  understood  from  the  following  illustrative  examples  and 
exercises. 

Example  I.  Find  an  equation  which  is  satisfied  by  the  following  pairs  of 
values  of  x  and  y  -. 

a;  =  12  3  4  5  6 

y  =  -i     1.5     1.22     1.125     1.08     LOG 

Plotting  the  given  pairs  of  values,  we  have  the  curve  in  Fig.  114.  We 
now  note  that  this  cui've  resembles  one  of  the  hyperbolic  curves  belonging  to 


Fig.  114 

the  system  y  =  Cj;",  but  with  this  difference ;  instead  of  approaching  the 
X-axis  as  an  asymptote,  it  apparently  approaches  the  line  ?/  =  1  as  an  asymp- 
tote.    We  therefore  assume,  as  a  trial  equation,  y  —  1  —  Cx".     If  the  given 


190 


LOCI   OP   HIGHER   ORDER 


[Chap.  IX. 


values  of  x  and  y  satisfy  this  equation,  then  the  values  of  \og{y  —  1)  and 
log  X  must  satisfy  the  equation 

log  (y  -  1)  =  log  C  4-  n  log  x, 

From  the  given  values  of  x  and  y,  we 


or  the  equation  of  a  straight  line. 

obtain, 

logx  =0         .301  .477 

log  (y  -  1)  =  .301     -  .301     -  .658 


.602 
.903 


.70  .80 

1.10     -1.22 


Plotting  these  values,  we  obtain  Fig.  115.     We  see  that  the  straight  line 
passing  through  the  first  and  next  the  last  of  these  points  very  nearly  passes 

through  the  others.  The  slope  of 
this  line  is  —  2  and  the  intercept 
on  the  F-axis  is  .301  =  log  2. 
Hence,  log  (y  —  1)  =  log  2  —  2 
log  X,   or 

2/  =  l+  V 
a;- 

By  actual  substitution,  this 
equation  is  seen  to  be  very  closely 
satisfied  by  the  given  pairs  of 
values  of  x  and  y. 

Logarithmic  paper  may  be  used 
in  this  example,  as  explained  in 
the  preceding  article. 

Example  II.  For  water  flow- 
ing in  pipes,  the  loss  of  pressure 
due  to  friction  is '  approximately 
propoi'tional  to  the  square  of  the 
velocity.  If  y  is  the  loss  of  pres- 
sure per  1000  feet  and  x  is  the 
velocity  in  feet  per  second,  then 
(approximately)  y  =  ax'^.  Find 
the  value  of  the  constant  a  so 
that  the  following  experimental 
data  will  fit  the  given  equation 
closely : 


Fig.  115 


1.9     2.8     3.6     4.3     4.8     6.1 
2        4        6        8      10      15 


7.2     8.2     9.1 
20      25      30 


Example  III.  Show  that  the  observed  data  in  the  preceding  example  will 
more  closely  satisfy  an  equation  of  the  type  y  =  ax".     See  Fig.  113. 

Example  IV.  The  following  observations  were  made  of  wind  pressure  on 
inclined  surfaces. 


Art.  139] 


EXAMPLES  AND   EXERCISES 


191 


Inclination  from  vertical :  30^  40°,  50°,  60°,  70°. 

Pressure  (pounds  per  square  foot)  :  5.5,  5.3,  4.4,  3.5,  2.1. 

DeteiTQine  the  curve  representing  the  pressure  as  a  function  of  the  angle. 

Suggestion.  Assume  the  equation  k  —  y  =  ax'K  Plot  the  observed 
values.  Estimate  k  =  6.  Plot  the  values  of  the  logarithms  of  Q  —  y  and  x, 
and  fit  a  straight  line  to  the  plotted  points. 


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Fig.  116 


Example  V.     In  a  certain  investigation  upon  the  strains  in  railway  bridges 
due  to  the  passage  of  trains,  the  following  data  were  found : 

x=      0    30    60    90    120    150    180    210    240    270    300    .330     360     390    420 
y^lOO    95    84    72      58      45      40      30      25      22      18      16       15       12      10 


192  LOCI   OF  HIGHER   ORDER  [Chap.  IX. 

Find  an  empirical  equation  which  these  observations  will  satisfy  with  close 
approximation. 

Suggestion.  Plot  the  given  pairs  of  values  carefully.  Note  that  the 
curve  obtained  seems  to  approach  the  X-axis  as  an  asymptote.  Since  the 
function  begins  with  a  maximum  value  at  x  =  0  and  steadily  decreases  in 
value  as  x  increases,  choose  type  7  as  a  trial  equation.     Plot  the  values  of 

i  and  x^  and  fit  a  straight  line  to  the  plotted  points. 

y 

Figure  116  shows  the  points  plotted  from  the  given  pairs  of  values  of  x  and  y 

and  also  the  locus  of  the  equation  y  =  — ~ —  in  which  a  =  2,000,000  and 

b  +  x^ 

b  —  20,000.     The  scales  are  indicated  on  the  figure. 

Example  VI.     In  a  series  of  experiments  on  the  adiabatic  expansion  for 

air,  the  following  data  were  obtained,  where  v  stands  for  volume  and  p  for 

the  corresponding  pressure. 

v=         3  4       5.2       G.O       7.3       8.5     10.0 

j;  =  107.3     71.5     49.5     40.5     30.8     24.9     19.8 

Find  the  empirical  equation  connecting  p  and  v. 

Suggestion.  Since  the  curve  obtained  from  the  given  pairs  of  values  of 
p  and  V  resembles  one  of  the  hyperbolas  of  the  system  y  =  Cx»,  plot  the  values 
of  log  p  and  log  v  and  fit  a  straight  line  to  the  plotted  points.  The  equation 
sought  ispv^-'^  =  497.7. 

EXERCISES 

1.  If  I  represents  the  length  of  a  steel  bar  and  t  represents  temperature, 
find  the  equation  connecting  I  and  t  from  the  following  observations  : 

/  =  1     1.0004     1.0008     1.0012     1.0016     1.0024     1.0040 
t  =  0  20  40  60  80  120  200 

2.  Find  the  equation  connecting  Q  and  h  from  the  following  set  of  obser- 
vations : 

h=    .583       .667       .750       .834       .876       .958 

Q  =  7.000     7.600     7.940     8.420     8.680     9.040 

3.  Show  that  the  following  set  of  corresponding  values  satisfies  an  equa- 
tion of  the  form  y  =  a6^.     Find  the  values  of  a  and  b. 

x  =  2.000       3.20        4.70         8.5       10.3         12.6 
?/  =  7.086     12.64     125.07     163.0     388.4     1178.0 

4.  The  following  set  of  observations  represents  the  deflection  d  of  a  beam 
of  length  L.     Find  the  equation  connecting  d  and  L. 

L=   12        16        20        24        28        32        36        40 
d=.17     .043     .085     .145     .220     .342     .512     .718 


Art.  140]  TYPE  y  =  a  +  bx  +  ex'-  +  dx^  +  ---  +  kx"-  193 

5.  Find  the  equation  connecting  u  and  v  from  tlie  following  set  of  corre- 
sponding values  : 

10=      .5     1.1     1.70     2.30     5.10     6.40 
V  =  13.6     4.0     2.37     1.84     1 33     1.28 

140.  Type  tj  =  a  +  bx  +  cx'^  +  dx^  +  ■■•  +  kx^.  When  a  given 
set  of  corresponding  values  will  not  satisfy,  in  a  satisfactory 
manner,  any  of  the  type-equations  1  to  7  (Art.  137),  the  general 

equation 

y  =  a-\-  hx -f  c.^•2  +  d.v^  -\ +  l-x" 

may  be  assumed.  By  substituting  pairs  of  corresponding  values 
in  this  equation,  the  values  of  the  constants  o,  b,  c,  •••A;  can  be 
determined  and  may  often  be  so  adjusted  that  the  locus  of  the 
resulting  equation  will  represent  the  function  to  a  fair  degree  of 
approximation  within  the  limits  of  observation. 

EXERCISES 

1.  Show  that  the  following  set  of  corresponding  values  satisfy  an  equation 
of  the  form  y  =  a  +  bx  +  cx'^.     Pind  the  values  of  a,  b,  c: 

x=    8    23     39     53    63 
y=10     10     27     33     36 

2.  Find  an  equation  of  the  form  y  =  a  +  bx  +  cx^  which  will  be  satisfied 
by  the  corresponding  values  of  angle  and  wind  pressure  in  Example  IV, 
Art.  139.  Why  is  the  equation  found  in  this  way  not  as  satisfactory  as  the 
equation  found  in  Example  IV  ? 


PAET    II 

SOLID   ANALYTIC    GEOMETRY 


^x 


CHAPTER   X 
SYSTEMS   OF   COORDINATES 

141.  Rectangular  and  oblique  coordinates.  As  has  been  said,  it 
requires  one  number  to  locate  a  point  on  a  line  and  two  numbers 
to  locate  a  point  in  a  plane  (Art.  4).  To  locate  a  point  in  space 
it  requires  three  numbers, 
called  the  coordinates  of  the 
point.  These  coordinates 
may  be  chosen  iii  several 
different  ways ;  any  par- 
ticular way  of  choosing 
them  gives  rise  to  a  system 
of  coordinates.  Thus  (Fig. 
117),  let  OX,  OY,  and  OZ 
be  three  linear  scales  hav- 
ing a  common  origin  0  and 
not  lying  in  the  same  plane. 
They  determine  in  pairs 
three  planes  XOT,  YOZ, 
and  XOZ,  called  the  coor- 
dinate planes.  If  through  any  point  in  space,  as  P,  three  planes 
are  drawn  parallel  to  the  coordinate  planes,  they  intersect  the 
linear  scales  in  the  points  D,  E,  and  F.  The  distances  x  =  OD, 
y  =  OJE,  and  z  =  OF  are  the  Cartesian  coordinates  of  the  point  P. 
The  linear  scales  OX,  OY,  and  OZ  are  called  the  X-,  Y-,  and  Z- 
axes,  respectively.  The  coordinate  planes  XO  Y,  XOZ,  and  YOZ 
are  called  the  XY-,  XZ-,  and  FZ-planes,  respectively.     The  sys- 

195 


196  SYSTEMS   OF   COORDINATES  [Chap.  X. 

tern  of  coordinates  thus  set  up  is  called  the  Cartesian  system  of 
coordinates. 

When  the  axes,  OX,  OY,  and  OZ  are  miitually  perpendicular, 
the  system  of  coordinates  is  called  rectangular  or  orthogonal.  If 
the  axes  are  not  mutually  perpendicular,  the  system  is  called 
oblique.  From  the  definition  of  the  coordinates  of  a  point,  and 
the  definition  of  a  linear  scale,  it  follows  that,  in  the  Cartesian 
system  of  coordinates,  to  each  point  in  space  there  corresponds 
one  set  of  values  of  x,  y,  z ;  and  to  each  set  of  values  of  x,  y,  z 
there  corresponds  one  point  in  space. 

EXERCISES 

(In  the  following  exercises,  take  the  axes  to  be  mutually  perpendicular. 
Cross-section  paper  may  be  used.) 

1.  Plot  to  scale  the  following  points,  the  coordinates  being  always  written 
in  the  order  (x,  y,  z): 

(1,  1,  1),  (2,  0,  3),  (-  4,  -  1,  ^  3),  (-  4,  2,  8),  (0,  0,  2),  (1,  -  3,  0). 

2.  Find  the  distance  between  the  points  (1,  —  2,  3)  and  (—  1,  2,  —  2). 

3.  Where  are  the  points  located  for  which  a;  =  0?  2/  =  0?  0  =  0?  What 
are  the  equations  of  the  coordinate  planes  ?  Where  are  tlie  points  located 
for  which  x  —  a  ;  y  —b  ;  z  =  c?  What  are  the  equations  of  the  planes 
parallel  to  the  coordinate  planes  ? 

4.  Where  are  the  points  located  for  which  x  =  0  and  y  =  0  ?  for  which 
X  =  a  and  y  =  b?  for  which  x  =  y?  for  which  x  =  y  —  z? 

5.  The  points  (2,  2,  3),  (2,  4,  3),  (4,  2,  3),  and  (3,  3,  2)  are  four  of  the 
vertices  of  a  parallelopipedon.  Find  the  coordinates  of  the  remaining  four 
vertices.     Is  there  more  than  one  solution  to  this  problem  ? 

142.  Spherical  coordinates.  Let  OX,  OY,  OZ  (Fig.  118)  be  a 
set  of  rectangular  axes,  and  P  any  point  in  space.  The  distance 
OP  =  r,  the  angle  ZOP  =  6,  and  the  angle  which  the  plaiie  ZOP 
makes  with  the  fixed  plane  XOZ  =  <^  are  the  spherical  coordinates 
of  the  point  P.     They  are  written  in  the  order  (r,  6,  (f>). 

If  the  point  P  is  on  the  surface  of  the  earth,  then  6  is  the  co- 
latitiide  and  <^  is  the  longitude  of  P.  If  P  is  on  the  celestial 
sphere,  then  0  is  the  co-declination  and  ^  the  right  ascension  of  P. 
If  Z  is  the  zenith,  then  6  is  the  zenith  distance  and  (^  is  the 
azimnth  of  P. 


Arts.  142,  143]    CYLINDRICAL  COORDINATES 


197 


143.    Cylindrical  coordinates.     In   Fig.   118,   let    OD=r',   the 

angle  XOD  =  (f>,  and  DP  =  z ;  (?•',  <^,  z)  are  the  cylindrical  coordi- 
nates of  P.     Again,  let  a, 
P,  y  denote  the  angles  which  2 

OP  =  r  makes  with  the  X-, 
Y-,  and  Z-axes,  respec- 
tively ;  then  (r,  «,  /3,  y)  are 
the  polar  coordinates  of  P. 

Spherical  coordinates  and 
cylindrical  coordinates  are 
modifications  of  polar  co- 
ordinates in  space.  Each 
is  in  common  nse  and  each 
has  its  advantages.  Spher- 
ical coordinates  are  espe- 
cially nseful  in  astronomy 
and  in  geodetic  surveying.  Fig.  118 


EXERCISES 

1.   Using  Fig.  118,  show  that  the  rectangular  coordinates  of  P  and  the 
spherical  coordinates  of  P  are  connected  by  the  following  formulas : 

x  =  r  s\nd  cos  0, 

y  =  r  sin  6  sin  0, 

z  =  r  cos  0. 


Conversely,  show  that 


r"^  =  x^  +  y-  -\-  z^, 
tan^g=^'  +  y', 


tan  (p  — 


2.  What  are  the  formulas  connecting  the  rectangular  coordinates  of  P 
with  the  cylindrical  coordinates  of  P  ? 

3.  Will  a  given  set  of  integral  or  fractional  values  of  r,  6,  4>  or  of  )-',  <^,  z 
locate  one  and  only  one  point  in  space  ?  Does  a  given  point  in  space  have 
more  than  one  set  of  polar  coordinates  ? 

4.  Locate  the  points  whose  spherical  coordinates  are  :  (3,  30°,  60°) , 
(2,  -,  TT  ],  (1,  45°,  45°).     Find  the  rectangular  coordinates  of  these  points. 


198  SYSTEMS   OF   COORDINATES  [Chap.  X. 

5.  Find  the  spherical  coordinates  and  also  the  cylindrical  coordinates  of 
the  following  points:   (2,  3,  4),  (3,  3,  -  2),  (-  1,  -  2,  1). 

6.  ^Vhere  are  the  points  located  for  which  r  —  const.  ?  for  which 
9  —  const.  ?  for  which  0  =  const.  ?  for  which  >■'  =  const.? 

7.  Where  are  the  points  located  for  which  0  —  const,  and  ^  =  const.  ?  for 
which  0  —  const,  and  r  —  const.  ?  for  which  r  —  const,  and  Q  =  const.  ?  for 
which  r'  —  const,  and  z  =  const.  ? 


CHAPTER   XI 


E, 


A 


DIRECTED   SEGMENTS  IN  SPACE 

144.  Projections  upon  the  coordinate  axes.  As  in  plane  geome- 
try, we  shall  call  a  segment  of  a  straight  line  to  which  a  direction 
has  been  attached,  a  directed  line-segment,  or  simply,  a  directed 
segment.     If   P1-P2   is   a 

directed    segment,    then  Z 

Pj  is  called  the  initial 
point,  and  P,?  the  termi- 
nal point. 

If  planes  are  drawn 
throngh  the  initial  and 
terminal  points  of  a  di- 
rected segment  and  per- 
pendicular to  each  of  the 
coordinate  axes  in  turn, 
these  planes  will  deter- 
mine upon  each  axis  a 
segment  called  the  pro- 
jection of  the  given  di- 
rected segment  upon  that  axis.  In  Fig.  119,  let  Pi=(xi,  2/1,  z^) 
and  P2  =  (x'2,  yz,  z^) ;  then  we  have 

projection  of  P1P2  upon  the  X-axis  =  x^  —  x^. 
projection  of  P1P2  upon  the  I^axis  =  2/2  —  Vi- 
projection  of  P1P2  upon  the  Z-axis  =  Z2  —  z^. 

145.  Length  of  segment.  A  segment  P1P2  is  the  diagonal  of  a 
rectangular  parallelopiped  whose  edges  are  the  projections  of  the 
segment  upon  the  coordinate  axes.     Hence,  we  have 


Fig.  119 


P,P,  =  ^{x,  -  x,f+  {y,  -  y,y+{z,  -  z,y. 
199 


200 


DIRECTED   SEGMENTS   IN   SPACE        [Chap.  XI. 


EXERCISES 

1.  Find  the  lengths  of  the  following  segments  and  their  projections  upon 
the  coordinate  axes. 

(a)  (1,  2,  3),  C- 2,  1,1);  (b)  (0,  0,  0),  (2,  0,  1)  ;  (c)  (3,  -  2,  0),  (2,  3,  1)  ; 
(d)  (0,4,1),  (-2,  -1,-2);     (e)   (0,3,0),  (3,  -1,0). 

2.  A  straight  line  five  units  in  length  has  one  extremity  at  the  origin  and 
is  equally  inclined  to  the  coordinate  axes.     Find  its  projections  upon  the  axes. 

3.  The  initial  point  of  a  directed  segment  is  at  the  point  (3,  2,-1)  and 
its  projections  upon  the  X-,  F-,  and  Z-axes  are  respectively  4,  —  6,  and  —  2. 
Find  the  coordinates  of  the  terminal  point  and  construct  the  figure. 

4.  If  the  terminal  point  of  a  directed  segment  is  (—1,  3,  5)  and  its  pro- 
jections upon  the  X-,  Y-,.  and  Z-axes  are  respectively  —  2,  3,  and  —  6,  what 
are  the  coordinates  of  the  initial  point  and  the  length  of  the  directed  segment? 

146.    Direction  angles  and  direction  cosines  of  a  directed  segment. 

The  angles  which  a  directed  segment  makes  with  the  positive 
directions  of  the  coordinate  axes  are  called  the  direction  angles  of 
the  segment.     The  cosines  of  the  direction  angles  are  called  the 

direction  cosines  of  the  segment. 
Through  Pi  draAV  lines  paral- 
lel to  the  axes  ;  i.e.  the  lines 
P,X',  P,Y',  P,Z'  (Fig.  120). 
The  direction  angles  of  P1P2 
are  then,  ^^_^.,p^p^^ 

^=Y'P,P„ 
y  =  Z'P^P^. 
If  I  is  the  length  of  P^P^, 
then  the  direction  cosines  are 
given  by  the  equations: 


^X 


COS  «  = 


I 


COS  /3 


_  ^2  -  yi 


Fig.  120 


COS  y 


147.    Relation  connecting  the  direction  cosines  of  a  segment. 
Theorem.    The  sum  of  the  squares  of  the  direction  cosines  of  any 
segment  is  equal  to  unity. 


Arts.  146,  147]     RELATION   CONNECTING  COSINES  201 

For  let  I  be  the  length  of  any  segment.    Then  (Art.  145),  we  have 
Z2  =  (a,'2  -  a."i)2  +  (?/2  -  2/i)2 + {z^  -  z^f. 

Dividing  by  l"^,  we  obtain 


I      J      \     I     J      \     I 

or  COS^  «  +  COS^  ^  +  COS^  y  =  1. 

EXERCISES 

1.  Find  the  length  and  the  direction  cosines  of  each  of  the  following 
segments : 

Pi=(4,  3,  -2).  P2=(-2,  1,-5);     Pi=(4,  7,    -2),  P2  =  (3,  5,    -4); 
Pi=(3,  -8,6),   Po=(6,  -4,6). 

2.  Find  the  lengths  and  the  direction  cosines  of  each  side  of  the  triangle 
whose  vertices  are  the  points  (.3,  2,  0),  (—  2,  5,  7),  and  (1,  —  3,  —  5),  the 
sides  being  taken  in  the  order  given. 

3.  Given  the  direction  cosines  of  the  segment  P1P2 ;  what  are  the  direc- 
tion cosines  of  the  segment  P2P1  ?  What  is  the  direction  of  a  segment  when 
cos  a  =0  ?  when  cos  j3  =0  ?  when  cos  7  =  0?  when  cos  a  —  cos  ^  —  0?  when 
cos  a  =  cos  7  =  0?  when  cos  ^  =  cos  7  =  0  ? 

4.  A  segment  is  five  units  long  and  its  initial  point  is  (—  2,  1,  —  .3).  If 
cos  «  =  J  and  cos  /3  =  |,  find  the  coordinates  of  the  terminal  point  and  the 
projections  upon  the  axes.  There  are  two  solutions,  find  each  of  them  and 
construct  the  figure. 

5.  Show  that  the  direction  cosines  of  each  of  the  lines  joining  the  points 
(4,  —  8,  6)  and  (—  2,  4,  -  3)  to  the  point  (12,  —  24,  18)  are  the  same.  How 
are  the  points  situated  ? 

6.  Find  the  direction  angles  of  the  segment  drawn  from  the  origin  to 
the  point  (8,  6,  0).     From  the  origin  to  the  point  (2,  —  1,  —  2). 

7.  Show  by  means  of  direction  cosines  that  the  three  points  (3,  —  2,  7), 
(6,  4,  —  2),  and  (5,  2,  1)  lie  on  a  straight  line. 

8.  If  two  of  the  direction  angles  of  a  segment  are  -  and  -  ,  what  is  the 
third  ?  ^  * 

9.  Show  that  the  numbers  3,  —  4,  and  —  2  are  proportional  to  the  di- 
rection cosines  of  the  segment  joining  the  origin  to  the  point  (3,  —  4,  —  2). 

10.  Show  that  any  three  real  numbers  a,  b,  and  c  are  proportional  to  the 
direction  cosines  of  the  segment  joining  the  origin  to  the  point  (a,  b,  c). 


202 


DIRECTED   SEGMENTS  IN   SPACE        [Chap.  XI. 


Fig.  121 


148.  Projection  of  a  segment  upon  any  line.     Let  PiP^  be  any 

segment,  and  AB  any  line  in  space.     Through  the  extremities  of 

the  segment  draw 
planes  perpendicu- 
lar to  AB.  These 
planes  determine  a 
segment  CD  upon 
AB  which  is  called 
the  projection  of 
PiPo  upon  AB. 
Through  P^  draw 

a  line  parallel  to  AB,  meeting  the  planes  in  the  points  Pj  and  Q. 

Let  6  represent  the  angle  P2P1Q  ;  then 

CD  =  P,Q  =  P,P2  cose. 

149.  Projection  of  a-  broken  line.  A.  series  of  segments  so  ar- 
ranged that  the  terminal  point  of  each  is  the  initial  point  of  the 
next  following  and  the  terminal  point  of  the  last  is  the  initial 
point  of  the  first,  constitutes  a  closed  line,  or  polygon,  in  space. 
The  sum  of  the  projections  of  the  sides  of  a  closed  line  upon  any 
line  in  space  is  clearly  equal  to  zero.  It  follows  from  the  fore- 
going property  that : 

Theorem.  The  sum  of  the  i^Tojections  of  a  series  of  segments 
joining  the  point  A  to  the  point  B,  upon  any  straight  line  in  space,  is 
equal  to  the  projection  of 

the  segment  AB  upon  that  Z  J'l 

line. 

For,  the  succession  of 
segments       AP^,      P1P2, 


P.P. 


BA     forms     a 


closed  line,  and  hence  the 

sum  of  the  projections  of 

its  sides  upon  any  line  is 

equal    to    zero;    i.e.    the 

sum  of  the  projections  of  Pj^  ^22 

the  sides  of  the  broken 

line  joining  J.  to  B  is  equal  to  the  projection  of  the  straight  line 

joining  A  to  B. 


Arts.  148-151]     PERPENDICULAR   SEGMENTS 


203 


150.  The  angle  between  two  segments.  When  two  segments 
do  not  intersect,  the  angle  between  them  is  defined  to  be  the 
angle  between  two  intersecting  segments  drawn  parallel  to,  and 
agreeing  in  direction  with,  the  given 
segments. 

To  find  the  angle  between  two 
given  segments  in  terms  of  their 
direction  angles,  let  li  and  h  be  the 
lengths,  and  aj,  ^i,  yj ;  Ojj  /?2>  72?  their 
respective  direction  angles.  From 
the  origin  draw  two  segments,  OP 
and  OQ,  having  lengths  and  direc- 
tion angles  equal  respectively  to  the 
lengths  and  direction  angles  of  the 
given  segments  (Fig.  123).  By  defi- 
nition, 6  =  POQ  is  the  angle  to  be 
found.  Let  the  coordinates  of  P  be 
X  =  OD,  y  =  DE,  and  z  =  EP.    Now, 

by  the  preceding  article,  the  projection  of  the  broken  line  ODEP 
upon  OQ  is  equal  to  the  projection  of  OP  upon  OQ.     That  is, 

?j  cos  0  =  X  cos  (u-\-y  cos  fSo  +  z  cos  y2. 
Dividing  through  by  /j  and  remembering  that  -  =  cos  a^,  etc.,  we 


Fig.  123 


have 


cos  e  =  cos  aj  COS  a.2  -|-  COS  Pj  COS  P2  +  COS  7i  cos  73. 


We  will  assume  that  the  angle  between  the  given  segments  is  the 
smallest  positive  angle  satisfying  this  equation  ;  that  is 

0   <    ^   <   TT. 


151.    Perpendicular  segments.     Parallel  segments. 

(a)   Tivo  segments  are  perpendicular  to  each  other  if    . 

cos  a,  cos  «2  +  cos  (3i  cos  jB^  +  cos  y^  cos  y2  =  0. 

For  then  cos  ^  =  0  and  therefore  6  =  90°. 

(6)   Tioo  segments  are  parallel  and  extend  in  the  same  direction  if 
their  direction  angles  are  equal,  each  to  each. 


204  DIRECTED   SEGMENTS   IN   SPACE        [Chap.  XL 

For  then  the  expression  cos  a^  cos  «2+cos  /3i  cos  ^82  +  003  y^  cos  72 
becomes  cos^  a^  +  cos^  /3i  +  cos^  yi  =  1  (Art.  147).  Therefore,  in 
this  case,  cos  6=1,  and  6  =  0°. 

(c)  Two  segments  are  parallel  and  in  opposite  directions  if  their 
angles  differ  by  180°,  each  from  each. 

^ovihen  cos«i=-cos«2, 

cos  /?!  =  —  COS  182, 

cos  yi  =  —  cos  y2. 

Hence  the  expression  cos  a^  cos  a2  +  cos  fi^  cos  IB2  +  cos  yi  cos  ya  be- 
comes —  (cos-  «!  +  cos^  y8i  +  cos^  yj)  =  —  1.  Therefore  cos  6=  —  l, 
and  6*  =  180°. 

EXERCISES 

1.  Find  the  angle  between  two  segments  whose  direction  cosines  are  as 
follows  : 

r n\     6        3        2     finfl      3       2        fi  .        (h'\      ^       2        1     n-nA     3       6       2.        /'/.N      2       _    1       2 

(^a)      7,      y,  y      ailU      y,     —   y,      y,  ^U  )       35—3,-3      di^U      y,      y,      y,  \^h )       3,  3,      3- 

rrn/l 3_     _4_      12 

""^  13'    13'    13' 

2.  Show  that  the  lines  whose  direction  cosines  are  f ,  ^,  f  ;  —  |-,  f ,  —  f  ; 
and  —  f ,  I,  f  are  mutually  perpendicular. 

3.  Show  that  the  points  having  the  coordinates  (—  6,  3,  2),  (3,  —  2,  4), 
(5,  7,  3),  and  (—  13,  17,  —  1)  are  the  vertices  of  a  trapezoid. 

4.  Show  that  the  points  (7,  3,  4),  (1,  0,  6),  and  (4,  5,  —2)  are  the  ver- 
tices of  a  right  triangle. 

5.  Show  that  the  points  (7,  2,  4),  (4,  -  4,  2),  (9,  -  1,  10),  and  (6,  -  7,  8) 
are  the  vertices  of  a  square. 

6.  Prove  that  if  the  direction  angles  of  two  segments  are  supplementary, 
each  to  each,  the  segments  are  parallel  and  in  opposite  directions. 

7.  Find  the  length  of  the  projection  of  the  segment  Pi=  (3,  2,  —  6), 
P2  =  (—  3,  5,  -  4)  upon  the  line  drawn  from  (1,  2,  3)  to  (3,  3,  1). 

8.  Find  the  length  of  the  projection  of  the  segment  Pi  =  (6,  3,  2), 
P2=  (4,  2,  0)  upon  the  line  drawn  from  (7,  —  6,  0)  to  (—  5,  —  2,  3). 

152.    Point  dividing  a  given  segment  in  a  given  ratio.     Let  P  be 

P  P 

a  point  on  the  segment  P^P^  situated  so   that  -^~=r,  a  given 

-/-to 

P  P  r 

number.     Then,  by  composition,  — —  = -.     Tlirough  P  draw 

planes  perpendicular  to  the  coordinate  axes.     These  planes  divide 


Art.  152]    POINT   DIVIDING  A  GIVEN  SEGMENT  205 

.Z        the  projections  upon  the  axes   in  exactly  the 
same  ratio  as  P  divides  the  segment;  that  is 

P,P  ^  D,D  ^  E,E  ^  F^F  ^      r 

P,P,~  D1D2     E,E.,     F,F,      r  +  1 

OD  =  X,  on,  =  x„ 

DiD^  =  X2  —  .^'l. 


Substituting,  we 
^X    have 


X^—  U/1   ~|~  v*^2  1  y 


Fig.  124 


'^1  "r  t'^2 

~    r+1    * 

Similarly,  v^e  obtain 


r  +  1 


r  +  1        r  +  1 


Z  =  Zi  +  (Zo  —  Zi) 


r     ^  gj  +  rz2 
r+1       r+1 


EXERCISES 

1.  Fiud  the  coordinates  of  the  point  dividing  the  segment  joining  the 
following  points  in  the  given  ratio  r. 

(a)   (3,  4,  2),  (7,  -  6,  4),  r  =  2.     (b)  (7,  3,  9),  (2,  1,  2),  r  =  4. 

2.  Sliow  that  the  coordinates  of  the  point  bisecting  the  segment 

(^.1,  ?/i,  ^i),  (3-2,  2/2i  •52)  aie  — - — ,    •- — —^-,    — 

3.  Find  tlie  coordinates  of  the  points  which  trisect  the  segment  (1,  —  2, 4), 
C-3,  4,  5). 

4.  Show  that  the  medians  of  the  triangle  whose  vertices  are  the  points 
(1,  1,  0),  (2,  -  1,  1),  and  (3,  2,  -  1)  meet  in  the  point  (2,  f,  0). 


206  DIRECTED   SEGMENTS   IN   SPACE        [Chap.  XI. 

5.  Show  that  the  medians  of  any  triangle  meet  in  a  point. 

Suggestion.  Let  the  coordinates  of  the  vertices  be  (xi,  ?/i,  Zi),  (0:2, 2/2,  ^2), 
and  (X3,  2/3,  Z2).     The  medians  meet  in  the  point 

a:i  +  a:2  +  Xz     ?/i  +  2/2  +  Vs     ^i  +  z.  +  zs 
3  '  3  '  3  ' 

This  point  is  the  center  of  gravity  of  the  triangle. 

6.  Show  that  the  lines  joining  the  middle  points  of  opposite  edges  of  a 
tetrahedron  pass  through  the  same  point  and  are  bisected  by  that  point. 

7.  Show  that  the  lines  joining  the  vertices  of  any  tetrahedron  to  the 
point  of  intersection  of  the  medians  of  the  opposite  face  meet  in  a  point 
which  is  three  fourths  of  the  distance  from  each  vertex  to  the  opposite  face. 
This  point  is  called  the  center  of  gravity  of  the  tetrahedron. 

8.  Find  the  ratio  in  which  the  point  (2,  —  1,  5)  divides  the  segment 
(4,  13,  3),  (3,  6,  4);  the  point  (2,  —2,  —6)  divides  the  segment 
(4,  -  5,  -  12),  (-2,  4,  6)  ;  the  point  (2,  1,  4)  divides  the  seg- 
ment (-3,  4,  2),  (7,  -2,  6). 


CHAPTER   XII 
LOCI  AND  THEIR  EQUATIONS 

153.  Surfaces  and  curves.  Iii  space  there  are  two  kinds  of 
loci  to  be  considered.  If  a  point  moves  according  to  a  given  law, 
it  will,  in  general,  describe  a  surface.  Thns,  if  a  point  moves  so 
as  to  be  always  at  a  given  distance  from  a  fixed  point,  it  will 
describe  a  sphere  whose  center  is  the  fixed  point  and  whose  radius 
is  the  given  distance. 

If  a  point  moves  so  as  to  satisfy  simultaneously  two  independ- 
ent laws,  it  will,  in  general,  describe  a  line,  straight  or  curved. 
Thus,  if  a  point  moves  so  as  to  be  at  a  fixed  distance  from  the 
point  A  and  at  the  same  time  at  a  fixed  distance  from  the  point 
B,  it  will  describe  the  circle  of  intersection  of  the  two  spheres 
whose  centers  are  at  A  and  B  and  whose  radii  are  the  given  fixed 
distances. 

154.  Equations  of  loci.  AVhen  the  law  governing  the  motion 
of  a  point  is  expressed  in  terms  of  the  coordinates  of  the  point, 
the  resulting  equation  is  called  the  equation  of  the  surface  de- 
scribed by  the  point.  The  surface  is  called  the  locus  of  the 
equation. 

Similarly,  when  a  moving  point  is  governed  by  two  independ- 
ent laws  and  these  laws  are  expressed  in  terms  of  the  coordi- 
nates of  the  moving  point,  the  resulting  equations  are  called  the 
equations  of  the  curve  described  by  the  point.  The  curve  is  called 
the  locus  of  the  equations. 

As  in  plane  geometry,  two  fundamental  problems  arise :  First, 
given  the  law  (or  laws)  governing  the  motion  of  a  point,  to  find 
the  equation  (or  equations)  of  the  locus ;  and  second,  given  the 
equation  (or  equations),  to  find  the  properties  of  the  locus.  These 
problems  will  be  illustrated  in  the  succeeding  pages. 

207 


208  LOCI  AND   THEIR  EQUATIONS         [Chap.  XII. 

155.  The  sphere.    Let  C=  (a,  b,  c) 

be  the  center  of  a  sphere  whose 
radius  is  r,  and  P=  (x,  y,  z),  any 
point  on  the  sphere.  .The  length 
of  CP  is  then, 

r   r  =  V(x  -  ay  -\-{y-  hf  +  {z  -  cf. 

Hence,  the  equation  of  the  sphere 
is 
^'^  Fxo.  125  (^^  -  «)'  +  iV-bf+i^-  of  =  r^^^ 

When  the  binominal  squares  are  expanded,  the  equation  has 
the  form  ^2  _^  ^2  ^  ^2  +  ^4.^  +  By  +  Cz  +  D  =  0,  (2) 

where  A,  B,   C,  and  D  are  constants  depending  upon  the  coor- 
dinates of  the  center  and  the  radius. 

Conversely,  an  equation  of  the  form  (2)  represents  a  sphere. 
For  it  can  be  written  in  the  form 

,  AW  f     ,  BW  f   ^  CV     A'B-'.C'      r) 

A   _B  _C 


2j       \        2j       \        2  J        4.       4: 
and  hence  represents  a  sphere  whose  center  is 


JA"^     B^      C^ 

and  whose  radius  is  a h  -r  +  ^; —  ^^• 

\  4        4        4 

The  sphere  is  real,  so  long  as  the  expression  under  the  radical 
is  positive ;  it  will  be  a  null-sphere,  or  a  point,  when  the  expres- 
sion under  the  radical  is  zero ;  and  it  will  be  an  imaginary  sphere 
when  the  expression  under  the  radical  is  negative. 

EXERCISES 

1.  Write  the  equation  of  a  sphere  whose  center  is  (5,  —  2,  3)  and  whose 
radius  is  1 ;  also  of  a  sphere  whose  center  is  (2,  —  3,  —  6)  and  which  passes 
through  the  origin.  What  is  the  equation  of  a  sphere  whose  center  is  on  the 
Z-axis,  has  the  radius  a,  and  passes  through  the  origin  ? 

2.  Which  of  the  following  spheres  are  real,  which  are  null-spheres,  and 
which  are  imaginary  spheres  ?    Find  the  center  and  radius  of  the  real  spheres. 


Arts.  155,  156]      SURFACES   OF   REVOLUTION 


209 


(a)  x"^  +  y^  +  z^  -2x  +  6y  -  8z  +  22  =0. 

(b)  x^  +  tf  +  z^  +  10  X  -  4  y  +  2  s  +  0  =  0. 

(c)  x^  +  y^  +  z^  +  4x  +  4y  +  6  z  +  I  =  0. 

(d)  x^  +  y^  +  z^  +  6x  =  0. 

(e)  x^  +  y"'  +  z^  +  4x  +  y  +  6 z  +  21  =  0. 

3.  Find  the  equation  of  the  sphere  passing  through  the  four  points 
(0,  0,  0),  (2,  8,  0),  (5,  0,  15),  (-  3,  8,  1). 

Suggestion.  Substitute  the  coordinates  of  the  given  points  in  equation 
(2)  and  solve  the  resulting  equations  for  the  unknown  coefficients  A,  B, 
CD. 

4.  Find  the  equation  of  the  sphere  passing  through  the  four  points 
(2,  5,  14),  (2,  10,  11),  (2,  5,  -  14),  (2,  -  10,  -^11). 

5.  Find  the  equation  of  each  of  the  two  spheres  whose  center  is  at  the 
origin  and  which  touch  the  sphere 

x2  +  y-2  +  ^-2  _  8a;-  6?/  +  242;  +  48  =  0. 

156.  Surfaces  of  revolution.  When  a  curve  in  tlie  XZ-plane  is 
rotated  about  the  JY-axis,  it  describes  a  surface  of  revolution. 
Every  point  on  the  curve,  as  Q, 
describes  a  circle  whose  plane  is 
perpendicular  to  the  X-axis  and 
whose  radius  is  the  ordinate  DQ. 

Let  the  coordinates  of  Q  be 
OD  =  X  and  DQ  =  z',  and  the 
equation  of  the  curve  MQR  be 
f{x,z')=Q.     Xow 


z'  =  DQ  =  DP  =  Vz^  +  9/. 

Hence,   we  have   the   following 
conclusion  : 


Fig.  126 


To  find  the  equation  of  the  sur- 
face described  by  MQR,  replace  z'  in  the  equation  f(x,  z')=  0  by  its 
value  ^/z^  +  y-. 

By  a  similar  consideration  we  may  find  the  eqviation  of  a  sur- 
face of  revolution  obtained  by  rotating  a  given  curve  about  either 
of  the  other  axes. 


210 


LOCI  AND   THEIR  EQUATIONS         [Chap.  XII. 


EXERCISES 

1.  Find  the  equation  of  the  ellipsoid  obtained  by  rotating  the  ellipse 
— I =  1  about  the  X-axis.     The  ellip.soid  of  revolution  is  called  the  pro- 

late  spheroid  when  a  >  &,  and  the  oblate  spheroid  when  a  <  6.  Explain, 
by  familiar  examples,  the  difference  in  form. 

2.  Find  the  equation  of  the  paraboloid  of  revolution  by  rotating  the 
parabola  z'~  =  ipx  about  the  X-axis. 

3.  If  the  hyperbola  - —  —  =  1  and  its  conjugate —  ——  1  are  ro- 

tated  about  the  X-axis,  how  will  the  two  surfaces  obtained  differ  ?  Find 
their  equations.  The  first  is  called  an  hyperboloid  of  two  sheets  and  the 
second,  an  hyperboloid  of  one  sheet. 

4.  Show  that  if  a  curve  in  the  XF-plane,  whose  equation  is  /(a;,  y)  =  0, 
is  rotated  about  the  X-axis,  the  equation  of  the  resulting  surface  is  found  by 
replacing  y  by  Vy'^  +  z^  ;  and  if  the  curve  is  rotated  about  the  F-axis,  the 
equation  of  the  resulting  surface  is  obtained  by  replacing  x  by  Vx"^  +  z'^. 

5.  What  is  the  equation  of  the  surface  obtained  by  rotating  the  parabola 
?/2  =:4^x  about  the  X-axis?  about  the  F-axis?  How  do  the  two  surfaces 
differ  ? 

157.  Cylinders.  If  a  straight  line  moves  so  as  to  be  always 
parallel  to  one  of  the  coordinate  axes  and,  at  the  same  time, 
intersects  a  curve  lying  in  the  plane  of 
the  other  two  axes,  it  describes  a  cyl- 
inder whose  equation  is  the  same  as  the 
equation  of  the  curve.  Eor,  suppose  the 
moving  line  is  always  parallel  to  the  Z- 
axis  and  meets  a  curve  in  the  XY-plane  ; 
then  the  x-  and  ^/-coordinates  of  any 
point  on  tliis  line  will  be  the  same  as  the 
X-  and  ^/-coordinates  of  the  point  where 
the  line  meets  the  curve,  and  will  conse- 
quently satisfy  the  equation  of  the 
curve  whatever  be  the  value  of  z.  Moreover,  the  x-  and  y-co- 
ordinates  of  a  point  not  on  the  cylinder  cannot  satisfy  the  equa- 
tion of  the  curve.  Therefore  the  equation  of  the  curve,  re- 
garded as  the  equation  of  a  locus  in  space,  represents  a  cylinder 
parallel   to  the  Z-axis.     Similarly  we  may  obtain   the  equation 


■^X 


Fig.  127 


Arts.  157-159]     PLANE   SECTIONS   OF   RIGHT   CONE  211 

of  a  cylinder  parallel  to  any  other  axis.     Hence,  we  have  the 
conclusion  : 

Any  equation  in  tivo  of  the  three  variables  x,  y,  z  represents  a 
cylinder  jiarallel  to  one  of  the  coordinate  axes. 

EXERCISES 

1.  The  following  equations  represent  loci  in  space.     Interpret  them  and 

draw  the  figures,     (a)  —  -f  ^  =  1 ;  (&)  z^  =  -^px;  {c)  s  +  3  «/  =  6. 
a-      t- 

2.  A  point  moves  so   as  to  satisfy  simultaneously  the  two   equations 

-  +  ^  =  1  and  ^  +  -  =  1.     Plot  its  locus  in  space. 
2      3  4      5 

3.  A  point  moves  so  as  to  satisfy  simultaneously  the  two  equations 
x^  +  ?/'^  =  4  and  -  +  -  =  1 .     Plot  its  locus  in  space. 

4.  Show  that  a  point  can  move  so  as  to  satisfy  simultaneously  the  three 
equations  3  x  +  2  ?/  =  6,  5  j/  +  4  0  =  20,  and  8  2—15  x  =  10. 

158.  The  right  circular  cone.  When  the  straight  line  z  =  mx 
is  revolved  about  the  X-axis,  it  generates  a  right  circular  cone 
whose  vertex  is  at  the  origin  and  whose  axis  is  the  X-axis.  Every 
generator  of  the  cone  makes  an  angle  with  the  axis  whose  tangent 
is  ni.     By  Art.  156,  the  equation  of  this  cone  is 

f/"2  +  s^  =  in-x'^. 

159.  Plane  sections  of  a  right  circular  cone.  Let  APB  (Fig.  128) 
be  the  curve  common  to  the  cone  and  any  plane,  as  AFPB.  In- 
scribe a  sphere  in  the  cone  touching  the  cutting  plane  at  F,  and 
the  cone  along  the  small  circle  LES.  The  cutting  plane  and  the 
plane  of  the  circle  meet  in  the  line  I)D^.  Through  P,  any  point 
of  the  ciirve  APB,  draw  the  generator  of  the  cone  VP,  meeting 
the  small  circle  in  E.  From  P  drop  the  perpendicular  PK  upon 
the  plane  LES,  and  draw  PR  perpendicular  to  DD^.  The  angle 
PRK=  a  is  the  angle  between  the  cutting  plane  and  the  plane 
LES  and  is  therefore  constant  for  all  positions  of  P.  The  angle 
PEK  =  (3  is  also  constant  for  all  positions  of  P.  The  lines  PF 
and  PE  are  equal  in  length,  since  they  are  tangents  to  the  sphere 
from  an  external  point.     Hence, 

PF^^PE^PK^  PK  ^  sina  . 
PR      PR      PR   '   PE       sin/3' 


212  LOCI  AND   THEIR  EQUATIONS         [Chap.  XII. 

V 


Fig.  128 

and  the  curve  APB  is  therefore  a  conic  having  one  focus  at  F, 
the  corresponding  directrix  being  Z)Z),  (Art.  94,  property  A). 

The  conic  will  be-  an  ellipse  when  a  <  y8,  a  parabola  when 
a  =  13;  i.e.  when  the  cutting  plane  is  parallel  to  one  of  the 
generators  of  the  cone,  and  an  hyperbola  when  a  >  ft.  In  the 
figure,  a  is  less  than  fi  and  the  section  of  the  cone  is  therefore  an 
ellipse. 

EXERCISES 

1.  The  equation  of  a  right  circulai-  cone  in  spherical  coordinates  is  ^=const. 
By  means  of  the  relations,  Art.  143,  exercise  1,  transform  this  equation  to 
rectangular  coordinates. 

2.  Rotate  the  straight  line  -  +  -  =  1  about  the  Z-axis  and  thus  obtain  the 

2      3 

equation  of  a  right  circular  cone  whose  vertex  is  at  the  point  (0,  0,  3). 

3.  From  Fig.  128,  show  how  to  locate  the  second  focus  of  the  section  of 
the  cone  and  its  corresponding  directrix. 

4.  The  cone  in  example  2  is  cut  by  a  plane  parallel  to  the  Y-axis  and 

meeting  the  XZ-plane  in  the  line  -  +  ?=  1.     Find  the  coordinates  of  the 
^  3      2 

foci  of  the  ellipse  which  this  plane  cuts  from  the  cone. 

5.  The  equation  —  +  ^ —  =  0  represents  a  right  circular  cone.     Write 

a^      (j2      j,2 

the  equation  of  the  straight  line  which  describes  this  cone  and  tell  about 
which  axis  it  is  revolved. 


CHAPTER   XIII 


THE  PLANE   AND  THE   STRAIGHT   LINE   IN  SPACE 

160.  The  normal  form  of  the  equation  of  a  plane.  Let  p  denote 
the  length  of  the  perpendicular  from  the  origin  to  the  plane,  and 
a,  (3,  y,  the  direction  angles  of  this  perpendicular.  If  L  is  the  foot 
of  the  perpendicular,  then  any  point,  as  P,  will  be  in  the  plane  if 
the  angle  OLF  is  a  right  angle ;  i.e.  if  the  projection  of  the  segment 
OF  upon  OL  is  equal  to  jk  But  the  projection  of  OF  upon  OL 
is  equal  to  the  projection  of  the 
broken  line  ODEF  upon  OL 
(Art.  149).  Leb  the  coordinates 
of  F  be  OD  =  x,  DE  =  y,  and 
EF  =  z ;     then 

a?  cos  a  -h  ?/  cos  P  +  s  cos  7  =  1?    (1) 

is  the  equation  sought. 
It  is  called  the  normal 
form  of  the  equation  be- 
cause it  is  expressed  in 
terms  of  the  perpendic- 
ular from  the  origin. 

It  follows  that  the 
equation  of  a  plane  is  of 
first  degree  in  the  vari- 
ables. 

We    shall    now    show 

that,  conversely,  every  equation  of  the  first  degree  in  the  variables 

X,  y,  z,  is  the  equation  of  a  plane. 

For,  let 

Ax  +  By  +  Cz  +  D  =  0  (2) 

be  any  equation  of  the  first  degree  in  x,  y,  and  z.     Now  if  the 
coordinates  of  a  point  F  satisfy  this  equation,  they  will  still  sat- 

"•21?, 


Fig.  129 


214  THE   PLANE  [Chap.  XIII. 

isfy  it  after  each  of  the  coefficients  A,  B,  C,  D  is  multiplied  by 
any  constant  k.  The  constant  Ti  can  be  chosen  so  that  the  equa- 
tion TcAx  -\-  JcBy  -{-  kCz  =  —  kD  will  coincide  with  equation  (1), 
term  for  term ;  that  is,  so  that 

kA  =  cos  a, 
kB  =  cos  ji, 
kC=  cos  y, 
kD  =  -2x 

Squaring  and  adding  the  first  three  of  these  equations,  we  have 
kXA"  +  £2 ^  (7-)=  1 ;  (Art.  147) 

and  therefore  k  = .     In    order  that    p  may   be 

positive,  the  sign  of  the  radical  must  be  opposite  to  the  sign  of  D. 
With  this  value  of  k,  equation  (2)  agrees  in  form  with  equation 
(1)  But  (1)  is  the  equation  of  a  plane;  therefore  (2)  is  the  equa- 
tion of  a  plane  for  which 

A 

cos  a  = 


cos/3  = 

cos  y  — 

P  = 


±  V^2  +  B'+C^ 

B 

±V.4'  +  5'+C2' 

C 

±  V^2  +  B^+  C^' 

-D 


The  distances  from  the  origin  to  the  points  where  a  plane 
meets  the  coordinate  axes  are  called  the  intercepts.  The  lines  in 
which  a  plane  meets  the  coordinate  planes  are  called  the  traces. 

EXERCISES 

1.    Construct  the  planes  and  find  their  equations,  for  which  {a)  tt  =-i 

|3=-,  7=^,i>=4;   (6)  «=  — ,  j8  =  5J:,  ^  =  Zi:,p=6;   (c)  cos  «  :  cos i3  :  cos  7 
00  .  o  4  o 

=  6  :  —  2  :  3,  j9  =  8  ;   (d)  cos  06 :  cos  ^  :  cos  7  =  —  2  :  —  1  :  —  2,  p  =  5. 

2     rind  the  equation  of  the  plane  such  that  the  foot  of  the  perpendicular 

from  the  origin  to  the  plane  is  the  point  (a)   (3,  —2,  6);     (6)  (2,  —5,  1) ; 

(c)   (3,4,  -2). 


Arts.  161-162]         EQUATION   OF   A  PLANE  215 

3.  Reduce  the  following  equations  to  normal  form  and  find  a,  ^,  7,  and  p. 

(a)  6x-3y  + 2  2-7=0.  (6)  x  -  V2  ?/ +  ^  +  8  =  0. 

(c)  X  -  4  ?/  -  2  0  -  3  =  0.  (fZ)  X  -  2  2/  -  3  =  0. 

4.  Find  the  intercepts  and  equations  of  the  traces  of  the  following  planes, 
(ff)  2  X  +  5  (/  -  3  g  -  4  =  0.       (6)  X  -  ?/  -  s  +  10  =  0.       (c)  3  x  -  ?/  +  s  =  0. 

5.  Find  the  area  of  the  triangle  which  the  coordinate  planes  cut  from  the 
plane  2  x  +  2  ?/  +  0  -  12  =  0. 

161.  Intercept  form  of  equation.  Let  the  x-,  y-,  and  2-intercepts 
of  a  plaue  be  a,  b,  and  c  respectively ;  then  (Fig.  126)  the  plane 
passes  through  the  three  points  A  =(a,  0,  0),  B=(0,  b,  0),  and 
C  =  (0,  0,  c).     Since  the  equation  of  the  plane  is  of  the  form 

Ax  +  By  +  Cz  +  D==0, 

this  equation  must  be  satisfied  by  the  coordinates  of  the  points 
A,  B,  and  C.     Hence,  ,         ^      „ 

Bb  +  D  =  0, 

Cc  +  D  =  0, 

from  which  A  =  —     ,  B  = ,  and  (7= .     Substituting  and 

a  b  c 

reducing,  the  required  equation  is 

^  +  i/  +  ^=l. 
a     b     c 

162.  Equation  of  a  plane  through  three  given  points.  If  a  plane 
is  required  to  pass  through  three  fixed  points,  the  coordinates  of 
these  points  must  satisfy  the  general  equation 

Ax  +  By  +  Cz+D  =  0, 

and  there  are  thus  three  equations  from  which  to  determine  three 
of  the  unknown  coefficients  A,  B,  C,  D  in  terms  of  the  fourth. 
Substituting  the  three  coefficients  thus  determined  in  the  general 
equation  gives  the  equation  of  the  plane  through  the  three  given 
points. 

EXERCISES 

1.    Write  the  equation  of  each  of  the  planes  having  the  following  inter- 
cepts and  find  the  length  of  the  perpendicular  from  the  origin  upon  each : 
(«)  3,  1,  2.     {h)    -1,-2,3.     (c)4,  -2,  5.     (d)   -  5,  2,  -  3. 


216  THE   PLANE  [Chap.  XIII. 

2.  Find  the  equation  of  the  plane  passing  through  the  points  (1,  0,  2), 
(0,  3,  4),  and  (—  1,  5,  0).  Find  the  intercepts  and  the  perpendicular  from 
the  origin. 

3.  Why  will  not  the  three  points  (1,  1,  2),  (3,  -  1,  3),  and  (5,  -  3,  4) 
determine  a  plane  ?  What  are  the  direction  cosines  of  the  segments  which 
join  the  first  point  to  each  of  the  other  two  ? 

4.  From  each  of  the  points  (2,  3,  0),  (-  2,  -  3,  4),  and  (0,  6,  0)  drop 
perpendiculars  to  the  XZ-plane.  What  are  the  coordinates  of  the  feet  of 
these  perpendiculars  ?  What  is  the  area  of  the  triangle  formed  in  the  XZ- 
plane  ?  Drop  perpendiculars  to  each  of  the  other  coordinate  planes  and 
compute  the  areas  of  the  triangles  formed  in  each.  These  triangles  are 
called  the  projections  of  the  space  triangle  upon  the  coordinate  planes. 

163.  Determinant  form  of  the  equation.  If  a  plane  is  required 
to  pass  through  three  given  points  (Xi,  y^,  Zi),  (x^,  y^,  z^,  and 
(.^•3,  ?/3,  z^,  the  general  equation 

Ax  +  5^/  +  C^  +  Z)  =  0  (1) 

must  be  satisfied  by  the  coordinates  of  these  points.  Hence  the 
following  equations  hold, 

Ax,  +  By,  +  Cz,  +  D  =  0,  (2) 

Ax^  +  By^  +  Cz,  +  D  =  0,  (3) 

Ax,  +  By,+  Cz,-\-D  =  0.  (4) 

But  in  order  that  the  four  equations  (1)  to  (4)  may  be  satisfied 
by  other  than  zero  values  of  A,  B,  C,  and  D,  it  is  necessary  and 
sufficient  that  the  determinant  of  their  coefficients  shall  vanish ; 
that  is,  we  must  have 


=  0. 


This  equation  is  of  the  first  degree  in  the  variables  x,  y,  z ;  and 
it  is  clearly  satisfied  by  the  coordinates  of  the  given  points. 
Therefore  it  is  the  equation  of  the  plane  passing  through  these 
points. 


X 

y 

z 

1 

x^ 

Vi 

Zi 

1 

*^2 

2/2 

^2 

1 

.^•3 

3/3 

h 

1 

Arts.  163,  164]    PERPENDICULAR  DISTANCE 


217 


EXERCISES 

1.  Using  the  determinant  form,  find  the  equation  of  the  plane  which 
passes  through  the  points  (2,  3,  0),  (—  2,  —  3,  4),  and  (0,  6,  0). 

2.  In  the  same  way,  find  the  equation  of  the  plane  passing  through  the 
points  (1,  1,  -  1),  (-  2,  -  2,  2),  and  (1,  -  1,  2). 

3.  Show  that  the  direction  cosines  of  the  normal  to  a  plane  passing 
through  three  given  points  are  proportional  to  the  cofactors  corresponding 
to  X,  y,  and  z  in  the  determinant  form  of  its  equation. 

4.  Show  that  the  cofactors  corresponding  to  x,  ?/,  and  z  are  proportional 
to  the  areas  of  the  projections  of  the  triangle  whose  vertices  are  (xi,  yi,  zi), 
(X2,  2/2,  ^2)5  and  (x^,  2/3,  03),  upon  the  coordinate  planes. 

164.  Perpendicular  distance  from  a  plane  to  a  point.  Given  the 
equation  of  the  plane  ABC  and  the  coordinates   of  the  point 


Fig.  130 


Pj  =  (xi,  yi,  Zi),  it  is  required  to  find  the  length  of  the  perpendic- 
ular DPi,  where  Z)  is  a  point  in  the  plane  ABC  and  P^  lies  outside 
of  this  plane.     If  the  equation  of  the  plane  is  not  in  the  normal 


218  THE   PLANE  [Chap.  XIII. 

form,  reduce  it  to  that  form  (Art.  160)  so  that  the  equation  is 

xcos  a  +  y  cos  (3  + z  cosy —2:>  =  0,  (1) 

where  the  direction  angles  and  the  perpendicular  are  known. 

Let  d  be  the  length  of  the  required  perpendicular,  so  that  the 

projection  of  OPi  upon  OJV  is  equal  to  jj  +  d.     But  this  projection 

is  equal  to  the  projection  of  the  broken  line  OEFP^  upon  ON. 

Hence 

'  p-\-d  =  x'l  cos  cc  +  ^1  cos  /3  -{-Zi  cos  y, 

or  (I  =  if  1  cos  a  +  yi  cos  p.+  si  cos  y—p. 

The  length  of  the  perpendicular  is  therefore  equal  to  the  result 
of  substituting  the  coordinates  of  the  given  point  in  the  left 
member  of  (1).  The  result  of  the  substitution  will  be  negative  if 
the  point  lies  on  the  same  side  of  the  plane  as  the  origin;  and 
positive  if  the  point  and  the  origin  are  on  opposite  sides  of  the 
plane. 

EXERCISES 

1.  Find  the  distance  from  the  plane  6x  —  By  +  2z  —  10  =  0  to  the  point 
(4,  2,  10).  From  the  plane  ix  +  3ij  +  12  z  +  6  =  0  to  the  point  (9,  -1,0). 
State  if  the  point  and  the  origin  are  on  the  same  side,  or  on  opposite  sides,  of 
the  plane. 

2.  Find  the  length  of  the  altitude  of  the  tetrahedron  from  the  vertex 
(2,  0,  1)  to  the  plane  of  the  vertices  (0,  5,  —  4),  (0,  3,  1),  and  (2,  -  7,  1). 

3.  The  X-  and  y-intercepts  of  a  plane  are  3  and  4,  respectively,  and  the 
plane  touches  a  sphere  whose  center  is  at  the  origin  and  whose  radius  is 
2.     Find  the  equation  of  the  plane. 

4.  Find  the  volume  of  the  tetrahedron  whose  vertex  is  the  point  (5,  5,  6) 
and  whose  base  is  the  triangle  cut  from  the  j)lane  x  +  2?/  +  5s— 10  =  0 
by  the  coordinate  planes. 

5.  Find  the  volume  of  the  tetrahedron  whose  vertices  are  (3,  4,  0), 
(4,  —  1,  0),  (1,  2,  0),  and  (6,  —  1,  4).  Of  the  tetrahedron  whose  vertices 
are  (3,  0,  0),  (0,  1,  0),  (0,  0,  6),  (5,  -  2,  4). 

6.  Find  the  locus  of  points  which  are  equally  distant  from  the  two  planes 
x  —  2y  -\-3z  -4^  =  0  and  2  x  +  Stj  —  z  -  5  =  0. 

7.  What  is  the  equation  of  the  locus  of  a  point  which  is  equally  distant 
from  the  origin  and  from  the  plane  x  +  y  +  s;  —  1=0? 

165.  Angle  between  two  planes.  The  angle  between  two  planes 
is  equivalent  to  the  angle  between  the  perpendiculars  to  these 


Arts.  165,  166]  PENCIL   OF   PLANES  219 

planes.     Let  the  equations  of  two  planes  be 

A^x  +  B^y  +  CiZ  +  A  =  0  and  A^x  +  B^y  +  C^z  +  D2  =  0. 

The  direction  cosines  of  the  perpendiculars  to  these  planes  are 
given  in  Art.  160.     Hence,  Art.  150,  we  have 

A,A,  +  B,B,  +  aC, 


cos  t^  = 


±VA'  +  A'  +  Ci2-  ±VA'  +  A'+C2=' 


the  signs  of  the  radicals  being  chosen  as  in  Art.  160. 

It  follows  from  this  formula  that :  two  planes  will  be  perpen- 
dicular if,  and  only  if, 

A,A,  +  B,B,  +  C\C,  =  0. 

For  only  then  can  cos  B  be  equal  to  zero,  and  consequently  9  =  90°. 
Two  planes  will  be  parallel  if,  and  only  if, 

A<i      Bi      L>2 

For  only  then  will  the  perpendiculars  to  the  planes  be  parallel  to 
each  other. 

EXERCISES 

1.  The  three  planes  a;  +  2/  +  2  —  2  =  0,  .r  —  ?/  —  22:  =  4,  and  2a5  +  ?/  —  5!  =  2 
meet  in  a  point  forming  a  trihedral  angle.  Find  the  vertex  of  the  angle  and 
the  three  dihedral  angles. 

2.  Find  the  equation  of  the  plane  which  passes  through  the  points 
(0,  3,  0)  and  (4,  0,  0)  and  is  perpendicular  to  the  plane  4a:  —  6x  —  s—  12  =  0. 

3.  Find  the  equation  of  the  plane  which  passes  through  the  point 
(1,  2,  4)  and  is  perpendicular  to  each  of  the  planes  2x  —  3y  —  s  +  2  =  0 
and  a;  —  ?/  +  2s  —  4  =  0. 

4.  Find  the  equation  of  the  plane  that  is  perpendicular  to  the  segment 
joining  (3,  4,  —  1)  to  (—  3,  6,  1)  at  its  middle  point. 

5.  Find  the  equation  of  the  plane  which  passes  through  the  point 
(3,  —  3,  0)  and  is  parallel  to  the  plane  3a;  —  jz  +  z  —  6  =  0. 

166.  Pencil  of  planes  with  a  common  axis.  The  system  of 
planes  passing  through  the  line  of  intersection  of  two  given 
planes 

A^x  +  B{ij  +  (7iZ  +  Di  =  0  and  A.x  +  B.jj  +  CaZ  +  i>2  =  0 


220  THE   PLANE  [Chap.  XIII. 

is  called,  a  pencil  of  planes  with  a  common  axis,  or  a  coaxial  pencil. 
The  pencil  is  represented  by  the  equation 

AiX  +  J5i2/  +C,z-{-D,  +  k  (Aox  +  Boy  +  C^z  +  D^)  =  0,     (1) 

where  k  is  an  arbitrary  constant.  Foi*,  it  is  clear  that  every 
point  whose  coordinates  satisfy  both  the  given  equations  will 
be  a  point  lying  on  the  locus  of  (1);  and,  since  (1)  is  of  first 
degree  in  the  variables,  it  is  the  equation  of  a  plane.  There- 
fore (1)  is  the  equation  of  a  plane  passing  through  the  line  of 
intersection  of  the  given  planes,  whatever  value  is  given  to  Jc. 

167.  Pencil  of  planes  with  a  common  vertex.  The  system  of 
planes  passing  through  the  point  Pi  =  (x^,  y^,  z^  is  called  a  pencil 
of  planes  with  a  common  vertex,  the  point  Pj  being  the  vertex.  It 
is  represented  by  the  equation 

A(x-x,)+B{y-y,)+C{z-z,)=0, 

where  A,  B,  and  G  are  arbitrary  constants.  For  this  equation  is 
the  equation  of  a  plane,  whatever  be  the  values  of  A,  B,  and  C, 
and  it  is  clearly  satisfied  by  the  coordinates  of  Pi.  Therefore  it 
represents  a  plane  passing  through  P^. 

EXERCISES 

1.  A  plane  passes  through  the  point  (3,  2,  —  1)  and  is  parallel  to  the 
plane  7x  —  y  +  z  —  li^O.    Find  its  equation. 

2.  Determine  k  so  that  the  plane  x  +  ky  — 2  z  — 9  =  0  shall  pass  through 
the  point  (1,  4,  —  3).  So  that  it  shall  be  parallel  to  the  plane  3  x  —  4  ?/  +  2  0 
—  5  =  0.    So  that  it  shall  be  perpendicular  to  the  plane  5x  —  3y  —  z  —  2  =  0. 

3.  Find  the  equation  of  the  plane  which  passes  through  the  intersection 
of  the  planes  2x  —  Sy  —  z  —  6  =  0  and  x  +  y  +  z  =  5,  and  (a)  passes  through 
the  point  (3,  —  2,  1);  (6)  is  perpendicular  to  the  plane  x  —  y  —  s  +  2  =  0. 

4.  Find  the  equations  of  the  planes  which  pass  through  the  intersection 
of  the  planes  x  —  y  —  3  z  —  4c  =  0  and  x  +  y  +  5z  —  6  =  0,  and  are  perpen- 
dicular respectively  to  each  of  the  coordinate  planes. 

5.  Find  the  equations  of  the  planes  which  are  parallel  to  the  plane 
6x—  5y  —  3z  —  2  =  0  and  which  touch  a  sphere  of  radius  3  whose  center  is 
at  the  origin. 

6.  Find  the  equation  of  the  plane  which  is  parallel  to  the  plane 
5x  —  3y  —  7z  —  8  =  0  and  such  that  the  point  (5,  —  1,  2)  lies  midway 
between  the  two  planes. 


Arts.  167,  168]     EQUATIONS  OF   A   STRAIGHT   LINE  221 

7.  Find  the  equation  of  a  plane  through  the  point  (2,  —  3,  0)  and  having 
the  same  trace  upon  the  XZ-plane  as  the  plane  x  —  3  y  -r  7  z  —  2  =  0. 

8.  Find  the  equation  of  the  plane  parallel  to  the  plane 

2x+y  +  2z-{-5  =  0, 
and  forming  a  tetrahedron  of  unit  volume  vdth  the  coordinate  planes. 

9.  Find  the  equation  of  the  plane  parallel  to  the  plane 

5x  +  Sy  +  z-7  -0 
such  that  the  sum  of  its  intei'cepts  is  23. 

10.  Find  the  equation  of  the  plane  having  the  trace  x  +  3  y  —  2  =  0,  and 
forming  a  tetrahedron  of  volume  f  with  the  coordinate  planes. 

168.  The  equations  of  a  straight  line  in  space.  If  Pi  =  (x^,  y^,  z{) 
and  P2  =  (x2,  3/2,  Z2)  are  any  two  points  in  space,  then  the  coor- 
dinates of  the  point  dividing  the   segment   P1P2   in   the   ratio 

^=r,  are  (Art.  152), 

r  +  1   ' 

y  =  y^,  (1) 

r  +  1 

When  r  is  allowed  to  vary,  these  equations  give  the  coordinates 
of  a  variable  point  on  the  line  Pj^P.,  and  are,  therefore,  the  para- 
metric equations  of  the  line  P1P2,  r  being  the  parameter. 

From  equations  (1),  or  from  the  figure,  Art.  152,  it  follows 

easily  that 

X  —  Xi  _  y  —  Vi       z  —  z^ 


^2      •^'1     Vi      Vi     ^2 


(2) 


These  equations  are  called  the  two-point  form  of  the  equations  of 
the  line  P1P2. 

Since  the  direction  cosines  of  P1P2  are  proportional  to  the  pro- 
jections  upon   the   coordinate   axes  (Art.    146),   we  have,  from 

equations  (2), 

X  -  a^i  ^  y-yi  ^  z-Zi  ^  .gx 

cos  a       cos  /8       cos  y 

These  equations  are  called  the  symmetric  form  of  the  equations  of 
the  line  P^Pi- 


222  THE   PLANE  [Chap.  XIII. 

As  an  example,  tlie  equations  of  the  straight  line  through  the 
points  (3,  —  2,  1)  and  (4,  5,  —  6)  are,  in  the  parametric  form, 

3+4 r  -2+5 r  1-6 r 

X  =  — ,  y  = ,  z  = 

r+1 ' ^  r+1     '  r+1 

In  the  two-point  form,  they  are 

1  7  -  7  ' 

The  direction  cosines  are  cos  a  =  — ^,  cos  jS  =  — ^,  cos  y  =  ~  ' 


V99  V99  V99 

EXERCISES 

1.  Find  the  equations  of- the  lines  joining  the  following  pairs  of  points : 
(a)   (0,  0,  -  2)  to  (3,  -  1,  0).  (6)  (-  1,  3,  2)  to  (2,  -  2,  4). 
(c)   (2,  -  3,  1)  to  (2,  -  3,  -  1). 

2.  In  the  preceding  exercise,  iind  the  coordinates  of  the  points  where 
each  line  meets  the  coordinate  planes. 

3.  rind  the  direction  cosines  of  each  of  the  lines  in  exercise  1. 

4.  Find  the  equations  of  the  line  through  the  point  (—  1,  2,  —  3)  if 

(o)    «  =  60°,  ^  =  60°,      7  =  45^ 

(6)    a  =  120°,         /3  =  60°,      7  =  135°. 

(c)  cos  a  =  ^  ,     cos  /3  =  i     cos  7  =  0. 

Show  that  the  given  values  are  possible  in  each  case  and  plot  the  line, 

5.  Find  the  equations  of  the  line  through  the  origin  and  equally  inclined 
to  the  axes. 

169.  The  projecting  planes  of  a  line.  The  planes  drawn  through 
a  given  line  and  perpendicular  to  each  of  the  coordinate  planes  in 
turn,  are  called  the  projecting  planes  of  the  line. 

The  equation  of  a  projecting  plane  can  contain  only  two  of 
the  three  variables  x,  y,  z  (Art.  157).  Hence  the  equations  of  the 
projecting  planes  can  be  found  from  equation  (2)  or  (3)  of  the 
preceding  article,  by  neglecting  one  of  the  ratios  involved.  Thus, 
for  example,  the  equation  of  the  projecting  plane  perpendicular  to 

the  XF-plane  is 

x  —  x^_y  —  y^ 

X^  -  .Ti       2/2  -  Vl 


Arts.  169,  170]    INTERSECTION   OF  TWO  PLANES 


223 


For  this  equation  represents  a  plane  parallel  to  the  Z-axis,  and  it 
is  satisfied  by  the  coordinates  of  the  points  Pj  and  P^-     Similarly, 
the  equations  of  the  other  projecting 
planes  can  be  found. 

The  equation  of  any  plane  through 
the  criven  line  is, 


=  Jc 


Fig.  131 


For,  this  equation  is  linear  in  x,  y, 
z  and  is  therefore  the  equation  of 
a  plane ;  moreover  it  is  satisfied 
by  the  coordinates  of  P^  and  Pa 
irrespective  of  the  value  of  the 
parameter  k. 


170.    The    intersection    of    two 
planes.     If  a  line  is  given  as  the 

intersection  of  two  planes,  its  equations  are  the  equations  of  the 
two  planes  considered  as  simultaneous  equations.  Thus,  the  equa- 
tions 

A,x  +  B,ii  +  C,z  +  Z>i  =  0  and  A^x  +  B.y  +  C^z  +  A  =  0       (1) 

are  the  equations  of  the  line  of  intersection  of  the  two  planes 
whose  equations  are  those  just  written.  The  direction  cosines  of 
the  perpendiculars  to  these  planes  are  respectively  proportional  to 
A^,  Pi,  Ci  and  A^,  B^,  C^ ;  and,  since  the  line  of  intersection  is  at 
right  angles  to  both  these  perpendiculars,  its  direction  cosines 
must  satisfy  the  two  equations 

A^  cos  a  +  Pi  cos  ^  +  Ci  cos  y  =  0, 

An  cos  «  +  P2  cos  ;8  +  C2  cos  y  =  0.     (Art.  151) 

Hence,  we  have 

cos  «  :  cos  ;8  :  cos  y  =  (P1C2  -  P2C1)  :  (A.,C,  -  A^CV)  :  (^liP,  -  A.B,). 

Therefore  the  equations  of  the  line  of  intersection  are 

X  —  x^       _       y  — 1/1        _        z  —z-^ 


Pi C2  -  PoCi     A.C,  -  A, a     A,B,  -  A.B, ' 
where  (a-j,  y^,  z^  is  any  point  on  the  line. 


224  THE   PLANE  [Chap.  XIII. 

To  find  a  point  on  the  line,  put  one  of  the  variables  equal  to 
zero  in  equations  (1)  and  solve  the  resulting  equations  for  the 
other  two. 

For  example,  consider  the  two  planes 

2x  +  iy  +  2z  -S  =0, 
and  5  X  +  6  ?/  +  ,j  -  16  =  0. 

The  direction  cosines  of  their  line  of  intersection  must  satisfy  the  two  equa- 

2  cos  «  +  4  cos  /3  +  2  cos  7  =  0, 
and  5  cos  a  +  6  cos  /3  +  cos  7  =  0. 

Hence,  cos  a  :  cos  ;8  :  cos  7  =  —  8  :  8  :  —  8  :  =  —  1  :  1  :  -  1. 

To  find  the  coordinates  of  a  point  on  the  line,  put  z  =0  in  the  equations  of 
the  planes  and  solve  the  resulting  equations  for  x  and  y.  In  this  way  we  find 
that  the  point  (2,  1,  0)  lies  on  the  line.     Therefore  the  equations  of  the  line 

are  •  01 

X  —  2  _  y  —  I  __    z 

~-  1  ~     1      ~  -  1' 

EXERCISES 

1.  What  are  the  equations  of  the  projecting  planes  of  the  line 

x  —  2_y  — 3_gt) 
6     ~      3      ~  2  ■ 

What  is  the  equation  of  the  plane  passing  through  this  line  and  through  the 
origin  ?     Through  this  line  and  through  the  point  (1,  —  2,  5)  ? 

2.  What  are  the  equations  of  the  line  through  the  point  (2,  5,  7)  if 
cos  a  =  |,  cos/3=  — ,  and  cos  7=0  ?  If  cos  «  =  ^,  cos/3  =  0,  and  cos  7= — '-  ? 
If  cos  «  =  0,  cos  ^  —  i,  and  cos  7  =  2 — -  ? 

o 

3.  Find  the  equations  of  the  projecting  planes  of  the  line 

2x-Sy  +  z-6  =  0,  x  +  y-3z-l  =  0 
by  eliminating  x,  y,  and  z  in  turn  from  these  equations. 

4.  Find  the  direction  cosines  of  the  line 

2x  +  3y-^2z-lS  =  0,  3x  +  6?/-3.?-24  =  0. 

5.  Find  the  coordinates  of  the  points  in  which  the  line 

2x  +  2y-Sz-2  =  0,     4x-?/-.s-6  =  0 
meets  the  coordinate  planes. 


Arts.  170,  171]   INTERSECTION  OF  LINE  WITH  PLANE    225 


6.  Reduce  the  equations  x  +  2y  +  6z  —  ^  =  0,  3a;— 2y—  10  s  —  7=0 
to  the  symmetric  form. 

Eliminating  in  turn  x  and  y  from  the  given  equations,  we  find  the  equa- 
tions of  two  of  the  projecting  planes  to  be 

2y  +  7  z  —  2  =  0  and  X  -  s  —  S  =  0. 

From  the  first,  s  —  — — ^^^^JZ — 2  j    and  from  the  second,  s  =  x  —  S.     Hence 

we  have, 

X  —  .^  _  y  —  1  _  s 

2      ~    -  7   ~2' 
from  whicli  the  symmetric  form  follows  at  once. 

7.  In  the  same  way,  reduce  the  equations 

2x  +  2y-Ss-2  =  0,    4:X  -  y  -  s  -  6  =  0, 
to  the  symmetric  form. 

8.  Reduce  the  equations  x  =  mz  +  a  and  y  =  nz  +  h  to  the  symmetric 
form. 

9.  Find  the  direction  cosines  of  the  following  lines  : 

(a)  4x-52/  +  32;  =  3,  4x-52/  +  0  +  9  =  O. 

ip)  2  .X  +  s:  +  5  =  0,  x  +  3  2;  -  5  =  0. 

(c)   3a;-?/-25;  =  0,  6x-32/-4s  +  9  =  0. 

10.  "What  are  the  equations  of  the  line  through  the  point  (2,  0,  —  2)  and 
perpendicular  to  the  lines 

^-^  ^y^  ^  +  1   and   ^  =  ^  +  ^  =  ^"  +  2  9 
2  12  3-1  2      ■ 

11.  What  are  the  equations  of  a  line  through  the  point  (2,  3,  4)  if 
cos  a  =  cos  /3  =  0  ? 

171.  Intersection  of  a  line  with  a  plane.  The  coordinates  of 
the  point  of  intersection  of  a  line  with  a  plane  must  satisfy  the 
equations  of  the  line  and  also  the  equation  of  the  plane.  Hence, 
to  find  these  coordinates,  solve  the  three  equations  simultaneously 
for  X,  y,  and  z. 

When  the  equations  of  the  line  are  not  given,  but  the  coordi- 
nates of  two  points  on  the  line  are  known,  a  more  expeditious 
method  is  to  write  the  equations  of  the  line  in  parametric  form 
(Ai't.  168,  (1)),  substitute  these  values  of  x,  y,  and  z  in  the  equa- 
tion of  the  plane,  and  solve  for  r,  thus  determining  the  ratio  in 
which  the  required  point  divides  the  segment  joining  the  given 
points.     For  example,  to   find  the  coordinates  of   the   point   in 


226  THE   PLANE  [Chap.  XIII. 

which  the  line  joining  the  points  (1,  —2,  0)  and  (3,  —4,  5)  meets 

the  plane  x  —  y-}-4:Z-\-2  —  0,  write  the  parametric  equations  of 

the  line ;  viz. :  ^       .. 

1+or 


x  = 

y  = 

and  z  = 


r  +  1' 
-  2  -  4  r 

r  +  1   ' 
5r 


and  substitute  these  values  of  x,  y,  and  z  in  the  equation  of  the 
plane.  In  this  way  we  find  r  =  —  ^.  Hence  the  point  of  inter- 
section is  (i|,  -  If,  -  If). 

The  line  whose  direction  angles  are  a,  ^,  and  y  will  be  parallel 
to  the  plane  Ax  -{-  By  +  Cz  +  D  =  0  if,  and  only  if, 

A  cos  a  -\-  B  cos  /3  +  C  cos  y  =  0. 

For  only  then  will  the  line  be  at  right  angles  to  every  perpen- 
dicular to  the  plane. 

The  line  will  be  perpendicular  to  the  plane  if,  and  only  if, 

^    ^    B    ^    C 

cos  a      cos  ^     cos  y 

For  only  then  will  the  line  be  parallel  to  every  perpendicular  to 
the  plane. 

EXERCISES 

1.  Find  the  coordinates  of  the  point  in  which  the  line  x  +  'ky  +  2 z  =  0, 
y  —  S  z  —  7  =0  meets  the  plane  3x  —  2y  +  s  +  4:=0;   the  coordinates  of 

the  point  m  which  the  line =  « =  - — -  meets  the  plane  x  -\- v 

3  2-4  f  -r  y 

+  ^  —  2  =  0  ;  the  coordinates  of  the  point  in  which  the  line  joining  the 
points  (2,  -  3,  1),  (2,  —  2,  4)  meets  the  plane  x  -  y  —  z  —  b=0. 

2.  Show  that  the  line  ^^  =  l^^  =  5  is  parallel  to  the  plane 

2  -7       3  ^ 

ix  +  2y  +  2z  =9. 

3.  Show  that  the  line  ^  =  1  =  ^  ig  perpendicular  to  the  plane 

o       ^       7 

Sx  +  2y-\-7z  =  8. 


Art.  171]     INTERSECTION   OF   LINE   WITH  PLANE  22? 

4.  Show  that  the  two  straight  lines  x  ~  2  =2  y  —  6  =  3  z  and  4  x  —  11 
=  4  ?/  —  13  =  3  2  meet  in  a  point.  Find  the  coordinates  of  this  point  and 
show  that  the  two  lines  lie  in  the  plane  2  x  —  6  y  +  3  z  +  14  =  0. 

5.  Find  the  equations  of  the  line  passing  through  (1,  —  6,  2)  and  per- 
pendicular to  the  plane  2x  —  y  +  6z=0. 

6.  Find  the  equations  of  the  line  passing  through  the  point  (—2,  3,  2) 
which  is  parallel  to  each  of  the  planes  3  x  —  y  +  z  —  0  and  x  —  z  =  0. 

7.  Show  that  the  six  planes,  each  containing  an  edge  of  a  tetrahedron 
and  bisecting  the  opposite  edge,  meet  in  a  point. 

8.  Prove  that  the  six  planes,  each  passing  through  the  middle  point  of 
one  edge  of  a  tetrahedron  and  perpendicular  to  the  opposite  edge,  meet  in  a 
point. 

9.  What  is  the  equation  of  a  plane  passing  through  the  point  (1,  3,  —  2) 
and  perpendicular  to  the  line 

X  —  3  _  y  —  i  _    z    c, 
2      ~      5      ~^  ■ 

10.  Find  the  equation  of  the  plane  determined  by  the  parallel  lines 

x  +  l^y-2  ^z    ^j^^^    x-3^y  +'i^z-l 
3  2  1  3  2  1 

11.  Find  the  equations  of  the  line  tangent  to  the  sphere  x-  +  y-  +  z-  =  9 
at  the  point  (2,-1,-2)  and  parallel  to  the  plane  x  -\-  3  y  —  5  z  —  1  =  0. 

12.  What  are  the  equations  of  the  line  passing  through  the  point  (xi,  j/i,  ^i) 
and  perpendicular  to  the  plane  Ax  +  By  +  Cz  +  D  =  0  ? 

13.  What  is  the    equation    of    the  plane    passing  through    the  point 
(xi,  yi,  Zi)  and  perpendicular  to  the  line 

X  —  Xi  _  y  —  J/2  _  Z  —  Z2  <> 

a  h  c 


CHAPTER   XIV 
EQUATIONS  AND  THEIR  LOCI 

172.  Second  fundamental  problem.  The  two  fundamental 
probleins  of  solid  analytic  geometry  were  stated  in  Art.  154. 
The  first  of  these  has  been  illustrated  in  the  preceding  articles  by 
finding  the  equations  of  certain  well-known  loci,  such  as  the 
sphere,  the  right  circular  cone,  the  plane,  the  straight  line.  In 
this  chapter  we  shall  consider  the  second  fundamental  problem  ; 
that  is,  given  an  equation  in  the  three  variables  x,  y,  z,  to  find  the 
form  and  properties  of  the  locus. 

1 73.  Construction  of  a  surface  from  its  equation.  The  following 
rules  serve  as  a  guide  in  sketching,  or  constructing,  a  surface 
from  its  equation. 

(1)  Symmetry.  If  the  equation  contains  only  even  powers  of 
one  of  the  variables,  the  surface  is  symmetrical  with  respect  to 
the  coordinate  plane  from  which  that  variable  is  measured.  For 
example,  if  the  equation  contains  only  even  powers  of  z,  and  the 
point  (a,  h,  c)  is  on  the  surface,  theii  the  point  (a,  h,  —  c)  will 
also  be  on  the  surface.  The  XF-plane  is  then  a  plane  of 
symmetry. 

If  the  equation  contains  only  even  powers  of  two  of  the  vari- 
ables, the  surface  is  symmetrical  with  respect  to  the  coordinate 
axis  along  which  the  third  variable  is  measured.  For  example, 
if  the  equation  contains  only  even  powers  of  y  and  z,  the  surface 
is  symmetrical  with  respect  to  the  XZ-plane  and  also  with  respect 
to  the  Xy-plane,  and  hence  with  respect  to  their  intersection,  or 
the  X-axis.     The  X-axis  is  then  a  line  of  symmetry. 

If  an  equation  contains  only  even  powers  of  all  three  of  the 
variables,  the  surface  is  symmetrical  with  respect  to  each  of  the 
coordinate  planes  and  therefore  with  respect  to  their  intersection, 
or  the  origin.     The  origin  is  then  a  point  of  symmetry. 

228 


Arts.  172-174]        THE   QUADRIC   SURFACES  229 

(2)  IntercejM.  The  length  of  the  segments  from  the  origin  to 
the  points  where  a  surface  meets  the  coordinate  axes  are  called 
its  intercepts.  These  are  found  by  putting  two  of  the  variables 
equal  to  zero  and  solving  the  resulting  equation  for  the  third 
variable. 

(3)  Traces.  The  sections  of  a  surface  made  by  the  coordinate 
planes  are  called  the  traces  of  the  surface.  The  equations  of  the 
traces  are  found  by  putting  each  variable  in  turn  equal  to  zero. 

(4)  Plane  sections  parallel  to  the  coordinate  planes.  The  equa- 
tion of  a  surface  and  the  equation  z=  Jc,  a,  constant,  are  together 
the  equations  of  the  curve  of  intersection  of  the  surface  with  a 
plane  parallel  to  the  XF-plane.  A  series  of  sections  parallel  to 
the  XF-plane  can  be  found  by  allowing  k  to  vary.  Similarly, 
sections  parallel  to  the  other  coordinate  planes  can  be  found. 

To  construct  a  surface,  it  is  customary  to  plot  the  traces  upon 
the  coordinate  planes  and  a  series  of  sections  parallel  to  at  least 
one  of  the  coordinate  planes. 

174.  The  quadric  surfaces,  or  conicoids.  The  locus  of  an  equa- 
tion of  the  second  degree  in  x,  y,  z  is  called  a  quadric  surface,  or 
conicoid.  It  can  be  shown  that  any  equation  of  the  second  degree 
in  X,  y,  z  is  reducible  by  a  proper  transformation  of  the  coordi- 
nate axes  to  one  or  the  other  of  the  two  forms 

Ax^  +  Bij^  +  Cz^  =  D,  (1) 

Aa>^  +  Bij^  =  2cz.  (2) 

If  the  coefficients  in  (1)  are  all  different  from  zero,  the  surface 
is  called  a  central  quadric,  the  origin  being  the  center.  By  the 
preceding  article  we  see  that  the  surface  is  symmetrical  with 
respect  to  each  of  the  coordinate  planes,  with  respect  to  each  of 
the  coordinate  axes,  and  with  respect  to  the  origin. 

If  the  coefficients  in  (2)  are  all  different  from  zero,  the  surface 
is  called  a  noncentral  quadric.  The  surface  is  clearly  symmetrical 
with  respect  to  the  XZ-  and  I"Z-planes  and  with  respect  to  the 
Z-axis,  but  it  is  not  symmetrical  with  respect  to  the  Xl'^plane 
nor  with  respect  to  the  X-  and  Y'-axes. 

If  one  or  more  of  the  coefficients  in  either  (1)  or  (2)  are  zero, 
the  surface  is  called  a  degenerate  quadric. 


230 


EQUATIONS  AND   THEIR   LOCI        [Chap.  XIV. 


175.    The  ellipsoid.     If  all  the  coefficients  in  (1)  of  the  preced- 
ing article  are  positive,  the  equation  can  be  written  in  the  form 


a"     b"     c- 


(1) 


and  the  surface  is  called  an  ellipsoid  (Fig.  132). 


Fig.  132 

The  intercepts  on  the  X-,  Y-,  Z-axes  are  respectively  ±  a,  ±  b, 
±  c.     The  numbers  a,  b,  c  are  the  lengths  of  the  semiaxes. 

The  traces  on  the  coordinate  planes  are  all  ellipses,  represented 
in  the  figure  by  ABCD,  BEDF,  and  AECF.  The  equations  of 
these  traces  are  respectively 

t  +  t  =  i^  -^^^'  =  1,  and  ^  +  5!  =  1. 


Equation  (1)  can  be  written  in  the  form 

3,2  y1 


+    ■ 


hHi-- 


=  1, 


from  which  we  see  that  any  section  of  the  ellipsoid  parallel  to 
the  XY^plane  is  an  ellipse  whose  semiaxes  are 


V^-S' 


and  b\\l 


Arts.  175,  176]     HYPERBOLOID   OF   ONE   SHEET  231 

Hence,  the  section  will  be  a  real  ellipse  only  vylien  z  is  confined  to 

the  range 

—  c<^  z^c, 

and  reduces  to  a  point  if  z  is  either  —  c  or  +  c. 

Similarly,  the  sections  parallel  to  the  I'Z-plane  will  be  real 
only  when  x  is  confined  to  the  range  —  a  <rc^  a,  and  the  sections 
parallel  to  the  XZ-plane  will  be  real  only  when  y  is  confined  to 
the  range  —h<y<h.  The  surface  is  therefore  wholly  inclosed 
within  the  parallelopiped  whose  edges  are  2  a,  2  b,  and  2  c. 

If  two  of  the  numbers  a,  h,  c  are  equal  to  each  other,  the  sur- 
face is  an  ellipsoid  of  revolution  (Art.  156 ;  Ex.  1) ;  and  if  all 
three  are  equal  to  the  same  number,  it  is  a  sphere  (Art.  155). 

EXERCISES 

1.  Construct  the  following  ellipsoids  : 

(a)  4.r2+9  2/"-+16sr-  =  144;  (b)  x^  +  W  y^-{-z'^  =  64  ;  (c)   16x^+y^+W  z'^=Qi. 

2.  Show  that 

i        ^        9        ^       16       ~ 

is  the  equation  of  an  ellipsoid  whose  center  is  (2,  1,  5). 

3.  In  general, 

(■'■  -  D-  ,  (y  -  m)^  ,  (^  -  ny  _ 

a^      "^        b'^  <fi 

is  the  equation  of  an  ellipsoid  whose  center  is  the  point  {I,  m,  n). 

4.  Show  that  a;'-^  +  2  y^  +  2  s:^  —  2  .i;  +  4  2/  -  8  z  +  10  =  0  is  the  equation  of 
an  ellipsoid  whose  center  is  the  point  (1,  —  1,  2)  and  whose  seraiaxes  are  1, 

,  and  —  •     Is  this  .surface  an  ellipsoid  of  revolution  ? 

2  2 

176.  The  hyperboloid  of  one  sheet.  If  two  of  the  coefficients, 
A,  B,  C  in  (1),  Art.  174,  are  positive  and  one  is  negative,  D  being 
positive,  the  surface  is  called  an  hyperboloid  of  one  sheet.  Suppose 
C  is  the  negative  coefficient,  the  equation  may  then  be  written  in 
the  form 

0  9  9 

a^     b-^     c^       ' 

from  which  we  see  that  the  intercepts  on  the  X-  and  I^-axes  are 


232 


EQUATIONS  AND   THEIR   LOCI        [Chap.  XIV. 


respectively  ±  a  and  ±  b ;  and  that  the  surface  does  not  meet  the 
Z-axis. 

The  trace  on  the  XY-^lane  is  the  ellipse 

a-     b~ 

while   the   traces    on   the   other  two  coordinate  planes  are  the 
hyperbolas 


=  1,  and  ^ 
52 


'-  =  1. 


Any  section  parallel  to  the  Xr"-plane  is  the  ellipse 


a' 


1+!-, 


+ 


b'   !+■ 


=  1, 


which  is  clearly  real  for  any  value  of  z  and  increases  indefinitely 

in  size  as  z  increases  or  diminishes  indefinitely. 

The  sections  parallel  to  the  YZ-plane  form  a  system  of  con- 
centric hyperbolas  (Art. 
108),  given  by  the  equa- 
tion 


=  1-^. 


Fig.  133 


If  a;  <  a,  the  transverse 
axis  of  the  correspond- 
ing hyperbola  is  paral- 
lel to  the  y-axis,  and 
if  x>a,  the  transverse 
axis  of  the  hyperbola  is 
parallel  to  the  .Z-axis- 
If  X  =  a,  the  section  of 
the  surface  is  the  two 
straight  lines 

0      c 


Similarly,  the  sections  parallel  to  the  XZ-plane  form  a  system  of 
concentric  hyperbolas.    The  form  of  the  surface  is  shown  in  Fig.  133. 


Art.  177]  HYPERBOLOID   OF   TWO  SHEETS  233 

EXERCISES 

1.  Construct  the  following  hyperboloids  : 

(a)  4  a;2  +  9  1/2 -  16  z'^  =  144  ;  (6)  x^  +  t  -  -^'^  =  25  ;   (c)  x'^  +  16  f-  -  z^=  64. 

2.  Show  that 


^_^  +  5!=iand-^  +  ^  +  ^  =  l 
a?     b-     C'  a'-      6'-      c^ 


are  the  equations  of  hjrperboloids  of  one  sheet ;  the  first  surrounding  the 
Y-axis,  and  the  second,  the  A'-axis. 

3.  ShoWthat         (,._iy2        (y-3)2        (z  -  2^  ^  ^ 

4  9  1 

is  the  equation  of  an  hyperboloid  of  one  sheet  whose  center  is  the  point 
(1,3,2). 

4.  Show  that        (x  -  ly      (y  -  m)-  _  (z  -  ny  _  j 

a^  62  (.2 

is  the  equation  of  an  hyperboloid  of  one  sheet  whose  center  is  the  point 
(I,  m,  n). 

5.  Show  that  3  x2  +  4  ?/2  —  ^2  _  g  ^  _  0  is  the  equation  of  an  hyperboloid 
of  one  sheet.     Find  the  coordinates  of  its  center. 

6.  Construct  the  surface  whose  equation  is 

x2  -  ?/2  +  2  2-2  -  6  X  +  2  y  +  4  s  +  9  =  0. 

7.  What  are  the  equations  of  the  planes  parallel  to  the  coordinate  planes 
which  cut  the  surface  9  x'^  —  ?/"  +  9  z-  =  36  in  pairs  of  straight  lines  ? 

177.  The  hyperboloid  of  two  sheets.  If  two  of  the  coefficients 
A,  B,  C  in  (1),  Art.  174,  are  negative  and  one  is  positive,  D  being 
positive,  the  surface  is  called  an  hyperboloid  of  two  sheets.  Sup- 
pose B  and  C  are  negative,  the  equation  can  then  be  written  in 
the  form 

O  i>  O 

a2      &2     c2       ' 

from  which  we  see  that  the  surface  does  not  meet  either  the 
y-axis  or  the  Z-axis  and  consequently  has  no  trace  upon  the 
YZ-^\sLne.  The  traces  upon  the  other  coordinate  planes  are 
hyperbolas. 


234 


EQUATIONS  AND   THEIR   LOCI        [Chap.  XIV. 


The  sections  of  the  surface  parallel  to  the   l"Z-plaue  form  a 
system  of  concentric  ellipses  given  by  the  equation 

b^      &      a^ 

from  which  we  see  that 
the  section  will  be  a  real 
ellipse  only  when  x  is  con- 
fined to  the  range 

—  a>  cc>  a. 

The  hyperboloid  of  two 
=  c.     The  form  of  the  siir- 


FiG.  134 


sheets  is  a  surface  of  revolution  if  h 
face  is  shown  in  Fig.  134. 


EXERCISES 

1.  Construct  the  hyperboloids  4  x^  —  9  2/^  —  16  0'2  =  1  and  x"^— 4  2/2  — 4  22  =  1. 
Which  is  a  surface  of  revolution  ? 

2.  Show  that 

_:^  +  r'_?'^land-^'-^  +  ?^=l 
a^      h'^      c?  «2     y2      f^2 

are  equations  of  hyperboloids  of  two  sheets  ;  the  first  surrounding  the  F-axis, 
and  the  second,  the  Z-axis. 

3.  Show  that  x'^  _  2  ?/2  _  4  j;^  -  2  a;  —  8  ?/  -  8  =  0  and  y"^  -  x'^  -  2  s"-  + 
6x  —  2?/  —  40  +  6=0  are  equations  of  hyperboloids  of  two  sheets.  Find  the 
coordinates  of  the  center  of  each. 


178.  The  elliptic  paraboloid.  If  the  coefficients  A  and  B  \\\ 
(2),  Art.  174,  have  the  same  sign,  the  surface  is  called  an  elliptic 
paraboloid.     The  equation  can  be  written  in  the  form 

where  c  may  be  either  positive  or  negative. 

The  trace  on  the  XF-plane  is  a  point,  namely,  the  origin  ;  while 
the  traces  on  the  other  coordinate  planes  are  the  parabolas 

/>j2  -1/2 

—  =  2  C2;  and  •—  =  2  cz. 


Arts.  178,  179]       HYPERBOLIC   PARABOLOID 


235 


The  sections  of  the  surface  parallel  to  the  Xy-plane  form  a 
system  of  concentric  ellipses  given  by  the  equation 


-\-^  =  ^cz, 


from  Avhich  we  see  that  the  section  will  be  real  only  when  c  and  z 
have  the  same  sign.  Hence  the  surface  lies  above  or  below  the 
Xl'-plane  according  as  c  is  positive  or  negative.  The  form  of 
the  surface  is  shown  in  Fig.  135, 
where  c  is  supposed  to  be  posi- 
tive. 

179.  The  hyperbolic  parabo- 
loid. If  the  coefficients  ^.l  and 
B  in  (2),  Art.  174,  are  opposite 
in  sign,  the  surface  is  called  a 
hyperbolic  paraboloid.  Its  equa- 
tion can  be  written  in  the  form 


IT- Of,. 
62~ 


The  trace  on  the  Xl'-plane  is 
here  a  pair  of  straight  lines 


>JC 


Fig.  135 


intersecting  at  the  origin,  and  the  sections  parallel  to  the  XY- 
plaue  form  a  system  of  concentric  hyperbolas  which  recede  from 
the  trace  on  the  XF-plane  as  z  increases  or  diminishes.  The 
transverse  axis  of  one  of  these  hyperbolas  is  parallel  to  the  X-axis 
if  c  and  z  have  the  same  sign,  and  parallel  to  the  F-axis  if  c  and 
z  have  opposite  signs. 

The  surface  is  saddle-shaped.  A  mountain  pass  between 
two  solitary  peaks  resembles  roughly  a  hyperbolic  paraboloid 
(Fig.  136). 

EXERCISES 


1.    Construct  the  following  surfaces 
(a)  X-  +  y'^  —  8  z. 
(c)  a;2  —  4  z^  =  16  y. 


(ft)  2/2 +  -2  =  4  a;. 
(d)  y-  ~  x^  =  10  z. 


236 


EQUATIONS  AND  THEIR  LOCI        [Chap.  XIV. 


2.  Reduce  each  of  the  equations  x^  +2y^  —  (Jx  +  4:y  +  Sz  +  11  =  0  and 
z^  —  Sy'^  —  4iX  +  2z  —  6y  +  l  =  0toa,  standard  form  and  determine  the  type 
of  paraboloid  of  which  each  is  the  equation. 

3.  A  point  moves  so  that  it  is  equidistant  from  two  nonintersecting 
straight  lines.     Show  that  its  locus  is  a  hyperbolic  paraboloid. 


Fig.  136 

4.    Discuss  the  equations  z  =  xy  and  z  = 
of  these  equations  ? 


x^  +xy  +  y'^.     What  are  the  loci 


180.    The  quadric  cone.     If  tlie  constant  term  D  in  (1),  Art.  174, 

is  zero  and  the  coefficients  A,  B,  C  are  not  all  of  the  same  sign, 

the  locus  of  the  equation  is  a  quadric  cone.     Suppose  that  C  is 

negative,   and  A  and   B  are   positive ;    the  equation   can   then 

be  written  in  the  form 

00"  ,  2/. 


a^     b'^     c^ 


(1) 


from  which  we  see  that  the  sections  parallel  to  the  X  Emplane  form 
a  system  of  concentric  ellipses  which  increase  in  size  indefinitely 
from  a  point  (when  2  =  0)  as  2  increases  or  decreases  indefinitely. 
Again,  if  P  =  (x^,  y^,  zj  is  any  point  whose  coordinates  satisfy 
(1),  we  can  prove  that  the  line  joining  the  origin  to  P  lies  entirely 
upon  the  surface.  For  the  coordinates  of  any  point  on  this  line 
are  clearly 


x  =  rx^,  y  =  ry^,  z. 


rz. 


Arts.  180-182 


PAIRS   OF   PLANES 


237 


where  r  is  any  number.  But  these  coordinates  satisfy  (1)  by 
virtue  of  the  hypothesis  that  the  coordinates  of  P  satisfy  (1). 
Therefore  tlie  cone  may  be 
generated  by  a  line  whicli 
rotates  around  the  origin  and 
intersects  an  ellipse  whose 
axes  are  parallel  to  the  X- 
and  y-axes,  Fig.  137. 

181.  Cylinders.  If  either 
(1)  or  (2),  Art.  174,  contains 
but  two  of  the  variables, 
the  corresponding  locus  is  a 
cylinder  (Art.  157).  The 
cylinders  are  therefore  de- 
generate quadrics. 

182.  Pairs  of  planes.     If 

an  equation  of  the  second 
degree  is  written  with  its 
right  member  equal  to  zero, 
and  its  left  member  is  then 
the  product  of  two  expressions  of  the  first  degree  in  the  variables, 
the  corresponding  locus  is  a  pair  of  planes.  For  the  equation  is 
satisfied  by  the  coordinates  of  any  point  which  render  either  fac- 
tor equal  to  zero.     Thus, 


Fig.  137 


is  the  equation  of  the  pair  of  planes 


^  +  ?^=Oand  ?-?/=0. 
a      b  a      b 


EXERCISES 

1.  Construct  the  cones  whose  equations  are 

(rt)  9  x2  -  36  if  -I-  4  ^2  =  0,  and  (5)   16  x^  -  4tf  -  z^  =  0. 

2.  If  in  (1),  Art.  174,  Z>  =  0  and  A,  B,  and  C  are  all  of  the  same  sign, 
what  is  the  locus  of  the  equation  ? 

3.  Show  that  x'^  +  iy'^  —  z'^  —  2  x  +  8  y  +  5  =  0  is  the  equation  of  a  cone 
whose  vertex  is  the  point  (1,  —  1,  0). 


238  EQUATIONS  AND   THEIR   LOCI        [Chap.  XIV. 

4.  In  general, 

a^  b  c^ 

is  the  equation  of  a  cone  whose  vertex  is  the  point  (?,  m,  n). 

5.  Construct  the  cone  whose  equation  is 

x2  +  2/2  —  2  ^  +  2  ?/  +  4  2  -  1  =  0. 

6.  Discuss  the  equations 

(a)  4  2/2  -  25  =  0  ;  (6)  2  2/^  +  5  s^  =  0  ;  (c)  2/^  -  a;2  -  4  2/  +  6  x  -  5  =  0. 

What  is  tlie  locus  of  each  equation  ? 

7.  Show  that  the  left  member  of 

3-2  _  2/--2  +  5;2  4-  2  xs  —  5  a;  —  2/  —  5  2  +  6  =  0 
is  divisible  hj  v  -\-  y  +  z  —  2.     What  is  the  locus  of  the  given  equation  ? 

183.  Ruled  surfaces.  If  a  straight  line  moves  according  to  a 
given  law,  it  describes,  or  generates,  a  ruled  surface.  Thus,  if  a 
line  moves  so  as  to  be  constantly  parallel  to  one  of  the  coordinate 
axes,  it  describes  a  cylinder  parallel  to  that  axis.  Again,  if  a 
line  rotates  about  a  fixed  point  and  intersects  a  fixed  curve,  it 
generates  a  cone  whose  vertex  is  the  fixed  point.  Cones  and 
cylinders  are  ruled  surfaces. 

If  the  equations  of  a  straight  line  contain  a  parameter  k,  then 
when  Ti  is  allowed  to  vary,  the  line  will  move  and  thus  describe  a 
ruled  surface.     For  example,  let  the  equations  of  the  line  be 

hx  +  y  -\-  z  —  k  =  0  and  x  —  hy  -\-  kz  -\-  1  —  {). 
From  the  first  of  these  equations  we  obtain 

k^'-l±^.  (1) 

1-x  ^  ^ 

and  from  the  second,  ?.  _  1  +  ^  (o\ 

~y  -z 

Therefore  the  coordinates  of  all  the  points  that  lie  on  the  line 
must  satisfy  the  equation 

y_±z^l^x^ 
1—xy—z 

or  a;2  ^  y^  -  z'- =  1,  (4) 


Arts.  183,  184]       EQUATION   OF   GENERATOR  239 

"whatever  the  value  of  Jc.  Conversely,  any  point  whose  coordi- 
nates satisfy  (4)  must  lie  on  the  given  line  for  some  value  of  k. 
For  (4)  is  equivalent  to  (3)  and  the  value  of  k  is  determined  from 
either  (1)  or  (2).  The  locus  of  (4)  is  an  hyperboloid  of  one 
sheet.  Therefore  the  given  line  generates  this  surface  when  k 
varies. 

EXERCISES 

1.  Find  the  equation  of  the  ruled  surface  generated  by  the  line  whose  equa- 
tions are 

x  +  y  =  kz,     X  -  ij  =  j- 

fC 

2.  Find  the  equation  of  the  ruled  surface  generated  by  the  line 

12/;  2      1      «i 

k 
(a)  when  —  =  2.   (/>)  ^Yhen  k  +  m  =  1,   (c)  when  km  =  3,  and  (d)  when  the 

m 

perpendiculars  from  the  origin  upon  the  two  planes  are  in' the  ratio  1  : 2. 

184.  Equation  of  generator.  It  frequently  happens  that  the 
equation  of  a  surface  indicates  at  once  that  it  is  a  ruled  sur- 
face.    For  example,  the  equation  (x -\-yy -\-(x-\- )j)z —  3  =  0  can 

be  written  ,  ,  ,  ^       r. 

(x  +  ij)(x  +  y  +  z)  =  3. 

o 

Hence  the  straight   line  x-\-y  =  -,  x-{-i/-\-z  =  k  lies  wholly  on 

ft 

the  surface,  whatever  value  is  given  to  k.    This  line  is  a  generator 

for  any  value  of  k. 

EXERCISES 

1.  Show  that  the  hyperboloid  of  one  sheet  is  a  ruled  surface. 
Suggestion.     The  equation  of  the  surface  can  be  written 

Hence  the  system  of  straight  line.s  (•^  +  ''\  =  k(l  +  -Y  ■{l—±\  =  l(i-'^ 
lies  wholly  on  the  surface. 

2.  Show  that  a  second  system  of  straight  lines  lies  wholly  on  the  hyper- 
boloid of  one  sheet. 

3.  Show  that  the  hyperbolic  paraboloid  is  a  ruled  surface,  having  two 
systems  of  straight  lines  lying  upon  it. 


240  EQUATIONS   AND   THEIR   LOCI        [Chap.  XIV. 

4.  Show  that  the  three  other  nondegenerate  conicoids  are  not  ruled 
surfaces. 

5.  Show,  by  the  method  of  this  section,  that  the  cone  —  +  ^  —  —  =  Ois 

n"^        tfi       r'^ 

a  ruled  surface.  a       o       c 

6.  Prove  that  y'^  —  ^ys +  4:z'^ +  xy —  ^xz^b  is  a  ruled  surface.  Are 
there  two  systems  of  lines  lying  upon  it  ?  What  is  the  form  of  the  surface  ? 
How  do  the  generators  lie  with  respect  to  each  other  ? 

185.  Tangent  lines  and  planes.  When  two  of  tlie  points  in 
which  a  straight  line  meets  a  surface  coincide  at  a  point  P,  the 
line  is  called  a  tangent  line  to  the  surface  and  P  is  called  its 
point  of  contact. 

In  general,  all  the  tangent  lines  at  P  lie  in  a  plane  called  the 
tangent  plane.     P  is  the  point  of  contact'  of  the  plane. 

To  find  the  equation  of  the  tangent  plane  at  a  given  point  on  a 

surface,  consider  the  ellipsoid  — \-~-\ —  =  1. 

a^     &^      c^ 

Let  Pi  =  (xi,  III,  Z])  and  P2  ^  (-''^'2)  Ihii  ^2)   be  any  two  points  in 

space.     The  equations  of  the  line  P1P2  are,  in  parametric  form 

(Art.  168),  x^+rx,  y^+ry,  z^±rz,  .^. 

r^l   '-^       r  +  1  '  r  +  1  '^  ^ 

To  find  the  coordinates  of  the  points  in  which  this  line  pierces  the 
ellipsoid,  substitute  the  values  of  x,  y,  and  z  in  the  equation  of  the 
surface  and  solve  for  r.     These  values  of  r,  when  placed  in  equa- 
tions (1),  give  the  coordinates  sought. 
The  equation  for  r  is  readily  found  to  be 


\^a^       b^      c^        J         \  ci^        V'        & 


Now  suppose  the  point  Pi  =  {x^,  y^,  z^  lies  on  the  ellipsoid.  The 
absolute  term  in  (2)  is  then  zero,  and  one  of  the  roots  is  i\  =  0,  as 
it  should  be.     The  other  root  is 

"V  a'         b^        c" 


a^       ¥      c- 


Art.  185]  TANGENT   LINES  241 

In  order  that  this  root  should  be  zero  also,  and  therefore  the 
two  points  of  intersection  coincide  at  Pj,  it  is  necessary  and  suffi- 
cient that  the  numerator  should  vanish.  Hence,  when  the  point 
P2  lies  on  the  plane 

a^-_^M-|_?i?_  1=0,  (3) 

a^       If       c^ 

the  line  P1P2  will  touch  the  surface  at  Pi ;  and  therefore  (3)  is 
the  equation  of  the  tangent  plane  at  P^. 

When  Pi  does  not  lie  on  the  surface,  equation  (3)  represents  a 
plane  called  the  polar  plane  of  Pj  with  respect  to  the  ellipsoid. 

A  line  perpendicular  to  the  tangent  plane  at  the  point  of  con- 
tact is  called  the  normal  to  the  surface  at  this  point. 

EXERCISES 

1.  Show  that  the  point  ('1,  2,  2^^  ^  Ues  on  the  ellipsoid  ^  +  l^  +  €i  =1. 

Find  the  equation  of  the  tangent  plane  at  this  point,  and  the  equations  of 
the  normal. 

2.  Derive  the  equation  of  a  tangent  plane  to  the  hyperboloid  of  one 
sheet. 

3.  Derive  the  equation  of  a  tangent  plane  to  the  hyperbolic  paraboloid. 

4.  Derive  the  equation  of  a  tangent  plane  to  the  quadric  cone  and  show 
that  any  tangent  plane  passes  through  the  vertex. 

5.  Show,  by  means  of  equation  (2),  Art.  185,  that  when  P2  lies  on  the 
polar  plane  of  Pi,  the  segment  P1P2  is  divided  externally  and  internally  in 
the  same  ratio  by  the  points  where  the  line  P1P2  meets  the  surface. 

6.  Show  that  the  length  of  a  tangent  line  to  a  sphere  from  the  point 
(xi,  2/1,  Zi)  is  equal  to  the  square  root  of  the  result  of  substituting  xi,  ?/i,  zi 
for  X,  y,  z  in  the  left  member  of  the  equation  of  the  sphere,  the  right  member 
being  zero. 

7.  Show  that  the  locus  of  points  from  which  tangents  of  equal  length 
may  be  drawn  to  two  spheres  is  a  plane.  This  plane  is  called  the  radical 
plane  of  the  two  spheres. 

8.  Prove  that  the  radical  planes  of  three  spheres  meet  in  a  line  called  the 
radical  axis  of  the  three  spheres. 

9.  Prove  that  the  radical  axis  of  three  spheres  is  perpendicular  to  the 
plane  of  their  centers. 

10.  Show  by  definition  that  the  tangent  plane  to  a  ruled  quadiic  contains 
the  two  generators  which  pass  through  the  point  of  contact. 


242  EQUATIONS  AND  THEIR  LOCI        [Chap.  XIV. 

186.    Circular   sections.     Certain  planes  cut  the  conicoids   in 
circles.     For  example,  consider  the  ellipsoid 

•         t  +  t  +  t^i  =  o. 
4      9      1 

The  coordinates  of  all  points  on  the  curve  of  intersection  of  this 
ellipsoid  Avith  the  sphere  x^  +  ?/^  +  z^  —  4  =  0  will  satisfy  the 
equation 

whatever  value  is  given  to  k.     When  k  =  \,  the  equation  becomes 

§^  _  ^'  =  0  (1) 

4        3(3  ^  ^ 

and  therefore  represents  two  planes.  Each  plane  cuts  the  sphere, 
and  therefore  the  ellipsoid,  in  a  circle. 

Any  plane  parallel  to  either  of  the  planes  (1)  cuts  the  ellipsoid 
in  a  circle.     For,  let 

2^  +  ^^_;b  =  0and2^-V— -m  =  0  (2) 

be  the  equations  of  any  two  planes  parallel  to  the  two  planes  (1). 
Combining  the  product  of  the  equations  (2)  with  the  equation 
of  the  ellipsoid,  it  is  easily  seen  that  all  points  common  to  the 
planes  (2)  and  the  ellipsoid  must  satisfy  the  equation 

— \-  —  -\ hkl  z ?/ 4-  m    z h  y —  km  —  1=0. 

444V    2        "^6/         V     2-^67 

But  this  is  the  equation  of  a  sphere  and  hence  the  planes  (2) 
meet  the  ellipsoid  in  circles.  There  are  thus  two  systems  of 
parallel  circular  sections,  each  being  parallel  to  one  of  the  planes 

EXERCISES 

1.  Find  the  equations  of  the  planes  which  cut  circles  from  the  ellipsoid 
9:fc'^  +  25?/2  +  169  02  ^  i. 

2.  For  what  values  of  k  and  m  will  the  equations  (2)  be  the  equations  of 
tangent  planes  to  the  ellipsoid  ? 


Arts.  186,  187] 


ASYMPTOTIC   CONES 


243 


3.   Find  the  equations  of  the  system  of  planes  which  cut  the  hyperboloid 


of  one  sheet  —  +  ^ —■ 

9      25      169 


1  in  circles. 


187.   Asymptotic  cones.     Consider  the  hyperboloid  of  one  sheet 

y'^  _  ^^  _  -1 


CI?     ¥ 


Let  Pj  =  (a*!,  y^,  z^  be  any  point  in  space.     The  equations  of  the 
line  joining  the  origin  to  Pi  are 

x  =  rx^,  y  =  ry„  z  =  rz^, 

?' being  a  parameter;  and  the  coordinates  of  the  points  in  which 

this  line  meets  the  hyperboloid  are  found  by  substituting  these 

values  of  x,  y,  and  z  in  the  equation  of  the  surface  and  solving  the 

resulting    equation    for    r. 

We  thus  obtain  for  r  the  ^ 

equation. 

1 


a- 


+f 


It  follows,  from  this 
equation,  that  as  Pi  is  made 
to  approach  the  cone 


;  + 


y 


-~=o, 


(1) 


the  values  of  r  increase  in 
absolute  value,  becoming  in- 
finite Avhen  Pj  lies  on  the 
cone.  Therefore  no  gener- 
ator of  the  cone  (1)  ever 
meets  the  hyperboloid. 
Moreover,  the  sections  of 
the  cone  and  the  hyper- 
boloid, parallel  to  the  XT- 

plane,  approach  coincidence  as  the  cutting  plane  recedes  from  the 
Xl^plane.     The  cone  (1)  is  called  the  asmyptotic  cone. 


Fig.  138 


244 


EQUATIONS  AND   THEIR  LOCI        [Chap.  XIV. 


EXERCISES 

1.  Find  the  e(juation  of  the  asymptotic  cone  of  tlie  hyperboloid  of  two 
sheets. 

2.  Show  that  tlie  asymptotic  cone  of  the  hyperbohc  paraboloid  consists  of 
two  planes. 

3.  Show  that  a  plane  determined  by  any  generator  of  the  hyperboloid  of 
one  sheet  and  the  center  is  tangent  to  the  as}^mptotic  cone. 

4.  Show   that   neither  the  ellipsoid  nor  the  elliptic  paraboloid  has  an 
asymptotic  cone. 

188.    Projecting  cylinders  of  a  curve  in  space.     The  cylinders 
whose  generators  intersect  a  given  space-curve  and  are  perpen- 


dicular to  one  of  the  coordinate  planes  are  called  the  projecting 
cylinders  of  the  curve. 


Arts.  188,  189]         PARAMETRIC   EQUATIONS  245 

To  find  the  equations  of  the  projecting  cylinders,  eliminate  x, 
y,  and  z  in  turn  from  the  equations  of  the  curve.  For  example, 
consider  the  curve  whose  equations  are 

a-2 -f- 7/2  ^  8  2,  a;2-?/2  =  4  2. 

Eliminating  x,  y,  and  z  in  turn  from  these  equations,  the  three 
projecting  cylinders  are  found  to  be 

y^  =  2z,  x^  =  6  z,  and  x"^  —  oy-  =  0. 

The  first  two  are  parabolic  cylinders,  shown  in  the  figure.  They 
intersect  in  the  given  curve.  The  third  equation  decomposes  into 
the  two  planes 

x  +  Vo  ^  =  0  and  x  —  Vo  y  =  0 

and  shows  that  the  given  curve  consists  of  two  parabolas  lying  in 
these  planes. 

EXERCISES 

1.  Construct  the  following  curves  : 

(a)  x'^  +  2/2  =  2.5,  y  +  ^  =  0.  (b)  X-  +  y-  —  ix  =  0,  X  +  y  +  z  =:  3. 

(c)  x^  —  if  =  i,  X  +  y  +  z  -  0. 

2.  Find  the  equations  of  the  projecting  cylinders  of  the  following  curves  : 

(a)  x^  +  y^-2y  =  0,  y'^  +  z^  =  4. 

(b)  2y-^  +  s^  +  ix-iz  =  0,  y-^  +  3z^  -8x  =  12z. 

(c)  x'^  +  tf  +  Z'  =  2.5,  x2  +4y2  —  z^:=  0. 
The  last  is  a  spherical  conic. 

3.  A  point  moves  so  as  to  be  constantly  2  units  from  the  Z-axis  and 
2  units  from  the  point  (2,  0,  0).  Find  the  equations  of  its  locus  and  plot  the 
curve. 

189.  Parametric  equations  of  curves  in  space.  If  the  coordi- 
nates of  a  point  in  space  are  each  functions  of  a  parameter,  the 
locus  of  the  point  is  a  line  in  space,  straight  or  curved.  For 
example,  the  equations  in  Art.  168  are  the  parametric  equations 
of  a  straight  line  in  space.     Again,  the  equations 

X  =  9-^,  y  =  r^,  z  =  r 

are  the  parametric  equations  of  a  curve  in  space.  The  equations 
of  the  projecting  cylinders  of  this  curve  are  found  by  eliminating 
r  from  each  pair  of  equations.     Thus,  the  projecting  cylinders  are 

x^  =  y^,  X  =  z^,  and  y  =  z"^. 


246 


EQUATIONS  AND   THEIR   LOCI        [Chap.  XIV. 


190.    The  circular  helix.     An  important  curve  in  mechanics  is 
the  circular  helix.     Its  parametric  equations  are 

X  =  a  cos  6,  y  =  a  sin  6,  z  =  hO, 

where  9  is  the  parameter. 

The  equations  of  the  projecting  cylinders  are 


x^  -\-  y^ 


a  cos  -,  y 


■    z 
a  sm  - 

h 


Hence  the  curve  lies  on  the  right  circular  cylinder  x^  +  y^  =  q2_ 

If  &  is  a  positive  number,  the  XZ- 
cylinder     stands     on     the     curve 

ic  =  acos-  or  ABCDE ;    and  the 
b 

yZ-cylinder  stands  on  the  curve 


y  =  a  sm 


The   helix   is  there- 


FiG.  140 


fore  a  curve  wound  around  the 
circular  cylinder,  the  distance  be- 
tween two  consecutive  turns  being 
2  67r  (Fig.  140). 


EXERCISES 

1.    Plot  the  curves  : 

(a)  x  =  2r,  y  =  r^,  z  =  —  —  ■      {b)  x 

^ecosi?,  y=6sm9,  g  =  -^  +  ^°^^^. 
4 


2.  Construct  a  circular  helix  when  6  is  a  negative  number. 

3.  Show  that  the  equations 

X  —  a  cos  d,  y  =  bsind,  z  =  md 

are  the  equations  of  a  helix  wound  on  an  elliptic  cylinder. 

4.  Construct  the  curve 

X  =  a  sec  0,  y  =  b  tan  9,  z  =  md. 

5.  The  three  equations 

Jcx  +  2y-\-  kz-2k  =  0, 

x  +  y  +  kz  —  Jc  =  0, 

kx  —  y  —  zk—1  —  0, 


Art.  190]  THE   CIRCULAR  HELIX  247 

are  the  equations  of  three  coaxial  pencils  of  planes  (Art.  152).  Express  the 
coordinates  of  the  point  of  intersection  of  the  three  planes,  for  any  value  of  k, 
in  terms  of  k  and  thiis  show  that  the  three  pencils  generate  a  space-curve. 
Construct  the  curve. 

6.    Shov?  that  the  point  (2,  1,  5)  lies  on  the  curve 

X' +  if  +  z'^  =  SO,  2/2=^, 

and  find :  (a)  the  equations  of  the  tangent  line  to  this  curve  at  the  given 
point,  (&)  the  equation  of  the  normal  plane  (perpendicular  to  the  tangent 
line)  at  the  given  point. 


ANSWERS 

Art.  3.     Page  10. 

1.    BA,  AB,  OB;  8,  8,  3  ;  No.  2.    73''  on  the  Fahrenheit  scale. 

4.   206°,  170^  5.    310^.  6.    115°. 

Art.  7.     Page  13. 

4.  P1P2  =  o,  P2P3  =  3,  P1P3  =  ^■ 

5.  P1P2  =  V37,  P2P3  =  3,  P1P3  =  i. 

6.  A  sqiiare,  one  side  4  units,  diagonal  4V2,  area  16. 

7.  (2,  0)  ;  each  V2  ;  each  2  ;  1. 

Art.  9.     Pages  15-16. 

1.  (2.598,  -  1.5)  ;   (2,  -  3.46-4)  ;   (-  1.25,  2.73). 

2.  (\/58,  -66°.8);  (2V5,  26°.6)  ;   (-  VM,  59°.03). 

4.  ?/  =  ±  3,  e  =  ±  36°.8. 

5.  (-171,0). 

9.   ^=(-8.66,  5)  or  (10,  60^).      P=  (0,  15)  or  (15,  0°).      C'=(10.39,6) 
or  (12,  -60°). 

Art.  11.     Pages  19-20. 

1.  (a)  5  and  2;  slope,  |.  (b)  1  and  -  9  ;  slope,  —  9. 
(c)    2  and  8  ;  slope,  4.  (d)  1  and  —  5  ;  slope,  —  5. 

2.  19°  6'  ;  40°  54'. 

3.  (fo)   26°  34';  75°  58' ;   146°  19'. 

Art.  12.     Page  21. 

1.    13.86.  2.    VT3.  3.    5.97. 

4.  12.73;  2.23;  14.92.  5.    (80, -),  144  miles. 

4 

Art.  15.     Page  24. 

1.    38°27'.  2.    11.402,  7.616,  7.211  ;  100°  30',  41°  3',  38° 27  . 

4.    1.792.  5.    10.  6.   35°  24'.    ' 

249 


250  ANSWERS 

Art.  17.     Pages  25-26. 

1.    (0,  3.5)  ;  (1,  -  1.5) ;  (3,  0).  5.    (0,  2)  and  (3,  3). 

6.    (1.125,  .25).  7.    (1,  0)  and  (0,  -2). 

Art.  19.     Page  28. 
1.    _6.  3.    -7.5.  4.    6.897.  6.    2.5. 

Art.  20.     Page  30. 
1.    88.5.  3.    22.935.  4.    151. 

Art.  21.     Pages  31-32. 

1.    122.5.  2.    25  acres  and  1  a  sq.  rd.  3.    60,294  sq.  ft. 

fx  =  -94.2         -27.8         188.0  54.1 

■    ^"^    \y=      66.0  157.3  68.3         -166.3 

(6)   N.  36°  E. ;  S.  67=^  36'  E. ;  S.  29°  46'  W. ;   N.  26°  48'  W. ;  N.  1°  12'  E. ; 
N.  50°  45'  W. 

(c)  42,277.06  sq.  ft. 

5.  150.5.  6.   300,  slope,  f. 

Art.  30.     Page  39. 

1.  Line  of  symmetry,  x  =  1  ;   intercepts  on  X-axis,  —  1  and  3 ;   intercept 
on  I^'-axis,   —3;  turning  point  (1,  —4). 

2.  Line  of  symmetry,  y  —  I;  intercept  on  X-axis,  | ;  intercept  on  F-axis,  1. 

3.  (±2,  ±2). 

6.  (_b_   iac-b-^\ 
\     2a        ia      } 

7.  Area  =  4  x  v'25  —  x^,  where  x  is  one  half  the  length  of  one  side.     Turn- 
ing point  for  x  =  — ^.     Max.  rect.  is  a  square  whose  side  is  5\/2. 

^  432 

8.  Number  of  sq.  ft.  of  lumber  is  — — \-  x^,  where  x  is  the  length  of  the  side 

x 
of  the  base.     Height  of  the  box  requiring  the  least  amount  of  lumber  is  3  feet. 

Art.  34.     Page  42. 

3.  (a)  y  =  a.         (b)   (x"-  +  y^  +  2  ax)'^  =  4  cfi{x'^  +  y'^)- 
(c)   {x^  +  y-^  —  ax)~  =  a2(x2  +  y^). 

4.  ?•  =  2  cos  d. 

Art.  41.     Page  50. 

9.  p»  =  4.  10.    I  =— ^  +  1. 

50000 

11.    ,^(93000000)^^     ^•  =  1.93,  nearly. 


ANSWERS  251 

Art.  45.     Page  56. 

1.  (a)  x-  +  y^-2y-S  =  0.  (b)  x^  +  2/2  +  4  x  =:  0. 

(c)  x^  +  y^  +  8  X  -  6  y  +  16  -  0.  (d)  x'^  +  ij-  -  2  x  -  i  y  -  SI  =  0. 

2.  x-  +  ?/2  -  4  X  —  6  2/  =  0.  3.    x2  +  y-  +  4  x  -6y  -IS  =  0. 

4.  x'-2  +  2/2  -  5  X  +  i  2/  -  I  =  0.  5.    10  X  -  8  ?/  +  3  =  0. 

6.  2(a  —  c)  X  +  2(6  -  d)y  —  6-  +  d"^  -  a-  +  c^  =  0. 

7.  x'^  +  2/2  =  16.  8.    x^  +  2/2  =  25. 
9.    x2  +  2/2  _  6  X  +  4  2/  -  12  =  0. 

Art.  46.     Page  57. 

1.  (a)  (3,  0)  ;  5.     (b)  (3,  -  2)^  3  V2".      (c)  (f ,  4) ;  J/,      (d)  (-  1,  2) ;  0. 
(e)   (4,0);  4.     (/)   (f,  J..)  ;  ii  V5.     (gr)   imag.      (h)  imag. 

2.  (19,  -Sg^-) ;  19.665,  nearly. 

Art.  48.     Page  59. 

1.  (a)x-y+l  =  0.  (b)  Sx-\-2y  +  l=0.         (c)  x  +  9  y  +  IS  =  0. 
(d)  7  X  +  4  2/  —  5  =  0. 

5.  (a)-l;|,|.         (6)|;3,_l.         (c)  3  ;  -  |,  2.         (d)    _  | ;  4,  |. 

Art.  50.     Pages  62-63. 

1     .  X-     ,    2/2  1  c, 

2.  ±  4,  0. 

^      25      9  ^   ^  36      9  ^  '^  25       25  ^   '^  100      50 

(e)  ^%l::^i. 
^   ^  50      25 


'161    ■  1  6  ■ 


4. 

5.  («)  a  =  V2,  &  =  V3,  c  =  iV3.  (6)  a  =  \/3,  /)  =  v'2,  c  =  ^\/3. 
(c)   «  =  V2,  6  =  A\/6,  c  =  V|.              (r/)  (1  =  1,  6  =  i\/2,  c  =  ^v^. 

6.  (a)  2Vi,      (6)   i|^,     (c)   fV2,    (d)   1. 

Art.  52.     Pages  65-66. 

J     x2_y2^j  2.   —--^  =  1.  3.    7.03+  and  1.03+. 

9       7  36     28 

4.  («)  3,  2;  1  Vis.     (6)  2,  3;  iVlS.    (c)  1,  4;  VT?.    (c7)  2\/to,  Vm;  iV5. 

5.  (a)    \/6,  2;  iViO.         (&)  4,  2;   V5.  (c)  Ap,    Vn;  a/^-^*- 

^m  ^     TO 

6.  (4a)  2f.    (4&)  9.    (4c)  32.    (id)  Vm.   (5a)  6.    (56)16.    (5c)-—"- 


252  ANSWERS 

Art.   53.     Page  67. 

1-    (1,0),  2.  2.    y2  =  8(x-l).  3.    0-2=  4(2/ -1). 

4.    (a)  y^  =  8x.     (b)  .r2  =  8  2/.  6.    (o)  8.     (5)  4.     (c)  6.     (d)  10. 

Art.  57.     Pages  70-71. 

1.  r^ -6  ?•  cos  (^- 60^)  =  7;       VaT. 

2.  ?•  cos  (e  -  (iO°)  =  5 ;       a;  +  ?/ VS  =  10. 

3.  >•  =  8  sin  e  ;     a;2  j^y2  -gy_ 

6.    r  =  ±  2  a  cos(6' —  45°)  ;      x- +  ?/-±rt.<:V2  ±  a?/ V^  =  0. 

Art.  59.     Pages  72-73. 

1.    r  = "^ .  2.    /■  = ^ .  5.    r  = 


4  —  3  cos  ^  3  cos  6  —  2  1  —  cos 

c~  —  a'^  ,  _  a-  —  c^ 


9.    r  =     '^  ~  " — .  10.   rr=     "   -^      .  12.    r  =      _     " . 

«  —  c  cos  ^  rt  +  0  cos  d  ^'2^  COS  e  —  1 

Art.   62.     Page  76. 

1       /„^    „       0     7        .)  /T  v'S  ...  Vl05        ,         \/165 

1.  (a)   a  =  6.  h  =  2,  c  =  V5,  e  = .  in)  a  = ,     b  = , 

_  '3  ^    '  7  11 

c  =  Vf 0,  e  =  V^j.     (c)  ffl  =  10,  &  =  5,  c  =  5 V5,  e  =  i V5.     (d)  a  =  2  Vff; 

6.  =  ^^,  c  =  f  Vl^,  e  ^  Vff.       (e)  rt  =  8,   &  =  5,  c  =  VSQ,  e  =  ^^. 

^  _  ,/Qq  ^ 

(/)rt  =  8,     6  =  5,    c=V89,    e=-^-5^. 

2.  2  2/ +x  =  0  and  2?/-x  =0.  3.    5i. 

4.    r  = ^ ,    6  =±  arc  tan  4. 

5  cos  6*  -  3  ' 

Art.  72.     Page  87. 

3.    a  =  2,  T=  ^^„-.     3:  =  2  cos  0  +  Vc-  —  4  siii-  <p  —  c,  where  c  is  the  length 
of  the  connecting  rod. 

Art.  78.     Page  94. 

1.    3  :)•'  +  7  !/'  =  10.  3.    .'•';/'  =  -  5.  4.    i/'^  =  a  V2  x'. 

Art.  81.     Page  97. 

01    5  ±  3y3\  _  ^     ,.  ^  ^  _  j2^  2,  ^  ,(4  _  ;2). 


3.    (4f,-3J).  .  8.    .  =  2(|y),,  =  2.(^ 

6.    2/  =  ±  ^^  a:.  9-    ;'■  =  a  cos*  6,  y  =  a  sin'*  ^. 


in  /1  («  +  ?))2sin2  6'    ,     ,  a  •       n 

10.    .'•  =  r\\  1  —  i — — — '- h  h  cos  d,  y  =  a  sm  ^. 


ANSWERS  253 

Art.  82.     Pages  98-99. 

2.    (1)^-|  =  1.  (-2)  2/  =  4(x+l).  (.3)  2/=-'2x  +  3. 

(4)  2/  +  5  X  +  10  =  0.        (5)  3  s:  +  2  y  -  6  =  0.  (6)  x  +  ?/  =  1. 

(7)  c^!/ +  .4Cx  =  ^5. 

Art.  83.     Page  100. 

2.  -  10  .*:  -  8  y  +  40  =  0.         —^x-^\y  +  i  =  0. 

3.  |x-3?/  +  9  =  0.         |a;-2?/  +  6  +  0. 

4.  10,  -  4,  -  6  or  -  15,  6,  9. 

Art.  84.     Page  101. 

1.  Ux-36y  =  0. 

2.  15  X  -  18  2/  -  320  =  0,  4  X  +  5  2/  -  3  =  0,  49  X  +  98  y  -  272  =  0. 

3.  3x  +  2y  +  1  =0.         5x-y  +  8  =  0. 

4.  xVS-y-  (V3-3)  =0.  5.    3y-10x-4  =  0. 

Art.  87.     Page  104. 

1.    (a)  x  +  yVS-10  =  0.     (b)  x  -  yVS +  10  =  0.     (c)  xV3-y +  10  =  0. 
(d)x  +  y  =  0.  (e)x  +  y^lV2.  (/)  x- ?/V3- 12  =  0. 

4.    ?/  =  5  and  21  X  +  20  */  -  145  =  0. 


Art.  88.     Page  105. 

1.   ^.  2.    2.35+.  3.    x2  +  2/-2  =  «25. 

4.  y-,  4  x2  +  4  ?/2  +  24  X  -  32  1/  -  69  =  0. 

5.  (a)  x2  +  2/2  =  5.  '  (6)  a;2  +  ^,2  +  8  a-  _  3  J,  ^  5,025  =  0. 

(c)  x2  +  2/2  +  (6  -  \/2)x  -  (4  -h  V2)2/  +  2/-  =  0. 

(d)  x2  +  2/2  +  3(2  -  V2)x  +  3(2  -  V2)y  +  ^(3  -  ^^2)  ^  ^ 

Art.  89.     Page  106. 

1.   45^.  2.    4°  4(3',  nearly.  3.    105°  15',  63°  26',  11°  19'. 

5.   2/  =-  f,  X  -  2/  =  f?,  17  X  +  7  2/  =  0.  __ 

8.    (a)  2x-32/  =  6.  (6)  3  x  ±  4  2/ =  15.  (c)  3  2/ -  2  x  =  5Vl3. 

Art.  90.     Page  107. 

1.    (1)  — +  ^  =  1.  (2)  il!  +  l^  =  l.  (3)^  +  ^=1. 

^^94  ^  M2      9  ^  ^  36      20 

(4)^  +  ^  =  1.  (5)^  +  ^=1.  (6)^  +  ^  =  1. 

^  ^  25      16  ^  ^  25      16  ^  ^   144      128 


254 

2.    (1)  a  =  5,  6  =  1,  c=iV26,  e  = 


ANSWERS 

V26 

(2)  rt  =  5,  b  =  \,  c  =  \/24,  e  = 

(3)  a  =  2,  6  =  3,  c  =  VIB,  e 


5 

V24 

5 

.  Vis 

2 

V5 


(4)  a  =  2,  6  =  3,  c=v'5,  e  = 

(5)  a  =  VlO,  6  =  2,  c  =  Vli,  e  =  V|. 

(6)  a  =  VlO,  6  =2,  c  =  V6,  e  =  Vf. 


3.   ^andl^. 
3  3 


4.   :^(9  +  2V5)  and  —  (9-2V5). 
6  6 


1.    —  +  ^  =  1.     }•  = 

9       4  3  -  V5  cos 


Art.  94.     Pages  110-111 
4 


„     a;'^      4w2      -, 
9        45 


15 


a;  =  3  cos  t,  y  =  2  sin  ^. 

3\/5 


—     X  =  3  sec  t,  y  =  "  "  "  tan  i. 
i  cos  ^  —  4  2 


27 


3.  ^+r  =  i.    ,.: 

36      27  6-3  cos  6 


4   ^_r  =  i     ,, 

■    16      25 
5.    c  =  4,  «  =  25. 
e       4 


25 


\/41  cos  ^  —  4 


=  6  cos  t,  y  =  3\/3  sin  t. 
X  —  4:  sec  t,  y  =  5  tan  ^. 


Art.  95.     Pages  113-115. 

1.    (a)  2/  =  i  X  +  2.         _(&)  2/  =-  fa:  ±  -2J.  (c)  ?/  =_  i  x  ±  VlO. 


(d)  y 


-4=  -^^^Q\x-3).  (e)  2/=_|a;±5. 


2.  (a)  2/  =  ±  f  X  ±  f.  (&)   2/  =  ±  X  ±  5.  (c)  ?/  =  ±  6  x  ±  7  V37. 

Art.  96.     Page  116. 

3.  (a)   (4,4).  (6)   (±-V-,  i-V^)^  (c)   (±|ViO,  ±  ^ VTO). 
^,^^12(-9T4V9l)^^9(lg|V91)y       (.^(i^,^.). 

I  Parabola,  (-2/,  ±  -i^o)  |  J  Ellipse,  (±  J/,  ±  |)        1 

*•    ^''^    1  Circle,  (-  f,  ±  H)    J  ■  ^^    1  Hyperbola,  (±  ^?,  ±  |)  } 


(c) 


^.     1       ,    ,  42V37      ,   7V37\ 

Circle,  (  ± ,  ± 

'  '  ""     '  37     / 


Ellipse, 


37 

/      .300  V37        13V37\ 
V         259     '         2-')9    J 


ANSWERS  255 

Art.  97.     Page  118. 

1.  3x  +  82/=:  19.  3.    3x  +  ?/  =  7.  4.    ?/ =  4,  y  =:  -  |  a;  +  i/. 
5.   X  +  2y  +  6  =  0.              6.    108^26'. 

10.    (a)2/  =  3x-4.  (6)  11 X- 2/ -  18  =  0.  {c)Sy  =  x  +  4. 

(d)  llx-2?/V'6-21  =  0. 

Art.  99.     Page  120. 

X  —  xi         2p     x—xi         o^xi 

3.  fVTS,    iV73,    5^,    |.        4.   27  2/ +  18  X- 88  =  0.         7.    8x-3?/  =  ]8. 

v^        ,^  ^      192  V65        1  ,       4v'65 

8.   ?)i  =---^  ;  subtangent  = ;  subnormal  = . 

3  65  3 

Art.  102.     Pages  124-125. 

2.  9y-2xy/S  =  0,     3?/  +  2xV'3  =  0. 

4.  2/iX-Xiy  =  0,      ^-^  =  0. 

« 
Art.  103.     Pages  126-127. 
1.    y  =  ix.  2.   2/ =-9.  3.   x  +  2j/  =  8. 

4.    13 x  + 11 2/ =  46.  5.    ?/  =  .r-l.  6.    y  =  ^x. 

Art.  104.     Pages  128-129. 

1.  (l)x-8?/  =  16.  (2)  x  +  2?/  +  6  =0.  (3)  5x  +  8?/  +  12  =  0. 
(4)  5x-62/  =  5. 

2.  (15,   -10).  3.    (-V"-,  ^).  4.    (-10,4). 
5     (^4^       i?5^\                      g     /       g^&'^xi                a^6'^j/i      \ 

■    i,       C  '        a/'  ■    Wyt^  +  b^xi^'    a^yi^  +  b-^xi'^j' 


Art.  109.     Page  138. 

1.   x2  +  2  2/2  =  10,     5  x2  -  120  2/2  =  24. 

Miscellaneous  Exercises.     Pages  138-140. 

1.    (2±3V6)x  +  2(6tV6)2/  =  40. 

2_    /149Tl2Vr5\  /-54V5  j:24V3\^^-^g_ 

o     n^    /4T6V6      6 -J,  V6 \  ,^.    /149t12V15      -  18V5  ±  8  VSX 


256  ANSWERS 

4.    VT5.  5.    ivTS.  6.    B2=zAC. 

10.    x  +  y  +  p  =  0.  11.    94,559,610  and  91,440,390. 

12.    arc  tan   ^^"^"  +  ^^  .  13.    500,000. 

m(«'^  —  b^) 

15.    ^  +  f^  =  i^-  17.   y  =  x(m  +  m'). 

a-      b^     a^ 

Art.   111.     Page  145. 

1.  (a)  Ellipse,  center  (|,  -  |),  foci  (|  ±  ^V95,  —  |). 
(6)  Ellipse,  center  (3,  -  J),  foci  (3  ±  |Vl7,^-  i). 

(c)  Hyperbola,  center  (^,  0),  foci  (J,  ±  tV3). 

(d)  Parabola,  vertex  (  —  2,  —  i^),  focus  (-2,  -  9). 

(e)  Hyperbola,  center  (0,  3),  foci  (0,  3  ±  2\/3). 
(/)  Parabola,  vertex  (-y-,  2),  focus  (5,  2). 

2.  (a)  Circle.  (6)  Hyperbola.  (c)  Two  lines  parallel  to  T-axis. 
{d)  One  line  parallel  to  X-axis.           (e)  Imaginary  lines. 

(/)  Two  lines  parallel  to  X-axis. 

Art.  112.     Page  148. 

1.  EUipse,  45°,  c  =  .577+. 

2.  (a)  Ellipse.  (6)  Imag.  ellipse.  (c)  Imag.  lines. 
{d)  Hyperbola.          (c)  Real  interesting  lines. 

3.  (05)3x2-1-2/2  =  2.  (&)  3x2-72/2  =  8.  (c)  7  x2  -  6  2/2  =  14. 
(d)  2x2-1-2/2  =  2.          (e)  s;2_  i2  2/2  =  _9. 

Art.   114.     Page  152. 

1.    (a)   (5x-2/-l)(x-f  2/  +  5)=0. 

(5)  ^^ y^ =  -  1;  6>  =  67°30';  center  (-  1,  0). 

2(\/2-l)      2(v'2-{-l) 

(c)   ^^ y^ =-  1 ;  61  =  67°  30'  ;  center  (-  1,  h). 

2(V2-1)      2(\/2-Hl) 

(d)  3  x2  -h  2/2  =  6 ;  61  =  45°  ;  center  (1,  -  2). 

(e)  ^^  ^■+  -J^—  =  1  ;  61  =  13°  18' ;  center  (1,  1). 
.3  +  V5      3  -  v'5 

(/)  Imaginary  lines. 

(gr)  — ^—3  -f-  — y—^  =  0  ;  61  =  arc  tan  2  ;  center  (3,  3). 
3  +  V5      3  -  \/5 


ANSWERS  257 

Art.   115.     Page  154. 


1.    (a)  ?/2  =  -  -|-x  ;  e  =  45°  ;  vertex  (^ —,  -  -^j 

nA  ,fi  -      ^^x;  e  =  4r)°  •  vertex  /  ^^^^     3V2\ 
(ft)  y   _  -  ^  i     4      '      8    j 

4V5  .     ,      „_„,.„  l./r .„„/4jv/5    4V5 

25   '     25 


(c)  .'(•-  =  —  - — -  a-  d  =  arc  sin  -  VS  ;  vertex  ( - 
^  '  25  5  V 


fd)  a;2  =  — 2^— ;  61  = -arc  tan—  ;  vertex  ( — ? — ,  - 

^  -  13VI3  2  5  V 13  Via  507 


6?/        „      1        ^      12           ^       /      2              2VI8 
•^ \  d  =  -  arc  tan  —  ;  vertex  ' 

!V13  2  5 

4  ?/                    .2  VS  / 

(e)  X-  = ~ ;  0  =arc  sm :  vertex 


Art.  116.     Page  155. 

1.    (ffl)  X  —  2/  —  1  =  0.  (&)  Imaginary  lines.  (c)  a;  +  ?/  ±  1  =  0. 

(d)  3  ..•  -  2  2/  +  2  =  0  and  3  a;  -  2  y  +  3  =  0. 


Art.  117.     Pages  156-157. 

1.  («)  Hyperbola.       ('))  Real  ellipse.       (c)  Parabola.       {d)  Hyperbola. 

2.  (a)  Parabola.  (ft)  Two  real  lines.  (c)  Real  ellipse. 
{d)  Two  real  lines. 

4.   k  =  \,  imaginary. 

7.    Vertex  (|,  4)  ;   focus  (§,  ^)  ;   directrix  4  ,r  +  2  ?/  =  7. 

Art.  118.     Page  159. 

1.  (rt)  3  X  +  8  ?/  -  5  =  0  and  3  .x  -  8  y  +  20  =  0.  {h)  x -\- y  =  4. 
(c)  ?/  +  4  =  0.  (d)  .'•  +  2  y  +  3  =  0.  (e)  8  x  -  37  ?/  +  18  =  0  and 
8  X  +  13  2/  -  18  =  0. 

2.  2/  —  4  X  =  8. 

Art.  119.     Page  162. 

I.    (5  ±\/l3)x  -  2(7  i  2V13);/  -  (13  ±  5VF^)  =  0. 
„      f  Axes,  X  +  y  =  Z  and  x  ~  y  =  9. 

I  Asymptotes,  (-  15  T  7\/5)x  ±  2  2/ Vs  +  15  ±  2  VS  =  0. 

3.  2/-x+l  =  0,     x  +  ?/-3  =  0,     x  +  ?/-l  =  0. 

Art.  121.     Pages  163-164. 

1.  17  X-  +  105  xy  —  48  2/^  +  210  x  +  39  =  0. 

2.  (rt)  5  X  +  5  2/  +  2  =  0.  (6)  3  X  -  2  2/  =  0.  (c)  6  x  +  1  =  0. 
(d)  ix-2ay  +  a^  —  c^  +  d-  =  4. 

4.  (i    -3). 


258  ANSWERS 

Art.  122.     Page  165. 

1.  x^  +  2  xy  +  y^  —  X  —  li  =  0  and  x^  —  2  xy  +  y'^  —  x  —  7  —  0. 

2,  x^  +  ixy  +  iy^  —  X  —  2  =  0  and  x^  —  i  xy  +  i  y"^  —  x  —  2  =  0. 

Art.  123.     Page  166. 

1.  (a)  x  +  ^  =  ±2{y-i),     Sx  +  l  =  ±iy-U),     x +  lb  =±(7  y  -  10). 

(b)  7  y  +  3(3  =  0,         3V7  ±  12  =  0. 

(c)  (2\/2±V5)x-(\/5tV2)2/=±(4v'5±2V2),    x-2y=0,    x+y=0. 

2.  (a)   (-5,  0)^  (-4,  3),    (3,  4),    p^,   -  |).       (&)   (±^,  -f)" 
(.)    (±1:^0^    ±?^0^,   (±2,   T2). 

Art.  124.     Page  167. 

1.  (a)  22  a;2  -  30  xy  +  7  2/2  -  22  X  +  9  2/  =  0. 

(&)  24  x2  -  73  xi/  +  29  y^  +  100  x  --  126  y  +  40  =  0. 

Miscellaneous  Exercises.     Pages  167-168. 

2.  (2,  -  3). 

4.  x2  +  2/2  -  18  X  -  36  ?/  +  81  =  0  and  x2  +  2/2  -  2  X  -  4  ?/  +  1  =  0. 

5.  (2,  0)  and  (5,  0).  ^  6.    (W,  -ft)- 

7.     (a)2/2=-|x.  (&)g_^=_l.  (c)  1+^=1. 

,,.     .,      IOV13  ,.  x2  2/2  , 

(d)  2/   = ^-  W   —  H —  1- 

13  _4(5+V5)      4(5 -v'5) 

(/)f-f=l.  (.<7)2/2=^x.  W./  =  2|lx. 

(0   (5  X  -  2  2/  +  3)  (5  X  -  2  2/  -  2)  =  0.  ( j)  x  -  3  2/  -  1  =  0. 

(A;)  x2  — 2/2  =  -10V2.        (?)  x2 -2/2  =— 40-        (?)i)  Imag.  lines, 
(n)  2/-^  =  -  AVt  a;-  (0)   (X  -  2  2/  -  2)  (X  +  2/  +  1)  =  0. 

Art.  145.     Page  200. 

2.    Each,  ^.  3.    (7,-4,-3).  4.    (1,  0,  11),  7. 

Art.  147.     Page  201. 

2.    Lengths  of  the  sides,  \/83,  V2r7^\/54. 
4.    Terminal  point,  (i,  |,  —  3  ±  f  V23). 

6.  Direction  cosines,  f ,  |,  0  ;  f ,  -  l    -  |.  8.   60°  or  120°. 

Art.  151.     Page  204. 

1.    (a)  90°.  (6)  arc  cos  —  /p  (c)  arc  cos  if. 

7.  4^.  8.    IJ3. 


ANSWERS  259 


Art.  152.     Pages  205-206. 

1.    (a)   (5f,  -2f,3i).  (6)   (3,  H,3|). 

3.    (-lf,2,  4|)and(-i,  0,  2^).  8.    -2.     J.     1. 

Art.  155.     Pages  208-209. 

1.  a;2  +  ?/2  +  ^2  _  10  a;  +  4 1/  -  6  0  +  37  =  0. 

x^  +  y^  +  z^  —  ix  +  6y  +  12  z  =  0.     x:^  +  7f  +  s"^  -  2  az  =  0. 

2.  (a)   (1,   -3,  4),  r  =  2.  (6)   (-5,  2,   -1),  r  =  5. 
(c)   (-2,  -2,  -3),  r  =  4.     (d)   (- 3,  0,  0),  r  =  3.     (e)  Imaginary. 

3.  x^  +  if  +  z^  -  2  X  -  8  y  -  16  z  =  0.  4.   x^  +  ?/2  _|_  ^2  _  4  ^  _  217. 
5.   a;2  +  ?/2  +  5;2  _  4  and  x^  +  y-  +  z-  =  •57(5. 

Art.  156.     Page  210. 

1.      ^  +  ^ +  £:=::  1.  2.     2/2  +  5-2::=  4  px. 

a2     ft2      52 
3.   ^'-2^'-^=l  and^'-^-^'  =  -l. 

a2        &2        ^-2  (jj2        52        ^2 

5.    2/2^02_4p;^_  J/-*  =  16jy2(x2  +  2;2). 

Art.  159.     Page  212. 

1.   x2  +  y2  =  ;s2  tan2  e.  2.   -+^-i^^-^  =  0. 

4       4  9 

4-    (rl'  0,  II)  and  (- H^  0?  Vs*)-  5.   6y  =  as.     About  the  Z-axis. 

Art.  160.     Pages  214-215. 

1.  (a)  xV2  +  y  +  z  =  8.  (b)  x  +  yV2  -  z  +  12  =  0- 
(c)  6x-2y  +  Sz  =  66.  (d)  2x  +  y  +■  2  z  +  16  =  0. 

2.  (a)  Sx-2y  +  6z  =  49.  (&)  2  x  -  5  ?/  +  0  =  30. 
(c)  Sx  +  4:y-2z+29. 

3.  ra)  fx-f2/  +  fa;  =  l.  (^b)  ^  i^  +  ^  y  -  1  z  =  i. 

,^v      1 ^^ 2    ^  _     3         .,.    _1_ 2_ §_2;-0 

\/21  V2I         V2I  V21  V14  '       Vli         Vli 

5.   54. 

Art.  162.     Pages  215-216. 

2.    16  X  +  6  ?/  -  5;  =  14. 

4.  Area  of  XZ-proj.  =  4.     Area  of  TZ-proj.  =  6.     Area  of  XF-proj.  =  12. 

Art.  163.    Page  217. 

1.   ox  +  2y +6z  =  12.  2.   x-3y-2z  =  0. 


260  ANSWERS 

Art.  164.     Page  218. 

1.   4.     -3.  2.    1.37,  nearly.  3.   ^■  +  ^-±?:^=l.         4.    58i 

3      4         12  ■* 

5.    8.     11.  6.   x  +  6y--iz-l  =  0. 

7.   x-2  +  y-  +  ,s2  —  (2/,?  +  a;.s  +  x?/)  +  X  +  ?/  +  .?  =  -i. 

Art.   165.     Page  219. 

1.  (4, -4,  2)  ;  118°7',  6r53',  G0°.  4.    3  x  -  ?/ -  g  +  5  =  0. 

2.  3  a;  +  4  y  -  12  ^  -  12  =  0.  5.    s  x  -  y  +  z  -  12  =  0. 

3.  7  a;  +  5  2/  -  s  -  13  =  0. 

Art.  167.     Pages  220-221. 

1.  7  X  —  ?/  +  2  —  18  =  0.  2.    -1- ;  impossible  ;  ^. 

3.  (a)   11  a; -4  2/ +  2  2; -43  =  0.     (&)  8  x  +  3  2/ +  5  z  -  36  =  0. 

4.  2/4-4^-1=0;  x+g-5  =  0;  4x-2/- 19  =  0. 

5.  6x-5?/-3;s±  6a/15  =  0.  8.   2  x  +  y  +  2  z  =  2-s/S. 

6.  5  .r  -  3  2/  -  7  ,s  -  20  =  0.  9.    5  x  +  3  2/  +  s  =  15. 

7.  x  +  1  z  =  2.  10.    3  X  +  9  2/  +  s  =  0. 

Art.  168.     Page  222. 

^    ^    3      -1  2  ^   ^       3  -5  2  ^  ^  '  •" 

2.  («)   (3,  -1,0),   (0,0,  -2),  (0,0,  -2). 

(5)  (-  4,  8,  0),  (I,  0,  V),  (0,  t,  f). 
(c)   (2,  —3,  0),  parallel,  parallel. 

3.  («)   4=,    -=^,    ^-     (6)^,    -^1,    ^-     (c)  0,0,1. 

V14      VU      Vli  \/38      V38      V38 

4.  («)  2(x  +  1)  =  2(2/  -  2)  =(^'  4  3)  V'2. 

(6)  -2(x  + l)  =  2(2/-2)  =  -(0+3)V2.     (c)  2  2/ -  x=  5,  s  +  3  =0. 

5.  x  =  y  =  z. 

Art.  170.     Pages  224-225. 

1.  X  -  2  2/  +  4  =  0,  ./■  -  3  ,5  -  2  =  0,  2  2/  -  3  0  -  0  =  0  ;  3  x  -  2 2/  -  6  2:=  0  ; 

25  X  -  32  2/  -  27  0  4-  46  =  0. 

2.  xV5  -  2  2/  =  2\/5  -  10,  s  =  7  ;   (x  -  2)  V3  =  2:  -  7,  2/  =  5  ; 
2(2/-  5)V2  =  ,s-7,  x  =  2. 

3.  5  27-72^4-4  =  0,  5x-8s-n  =  0,  7x-8  2/-19  =  0. 

4.  ^,    0,    -1-. 

\/2  V2  ■ 

5-   a  -iO),  (!,  0,  f),  (0,  -J^,-V)- 


ANSWERS  261 

1  1         a 

3  3  3 

X  —  a  y  —  b  z 

m  n  1 


Vwi-  +  n-  +  1      y/m-  +  n^  +  1      Vm'-^  +  n^  +  1 

9.    («)  -|^,  ^,0.  (&)  0,1,0.  (c)-=-|,o,  ^. 

V29     \/29  Vis  Via 

10.   51=1^  =  ^  =  ^^.  11.   a;  =  2,y  =  3. 

4         2-5  '  ^ 

Art.   171.     Pages  226-227. 

1.    (0,  1,  -2);   (-10,  -7,  Ifl);    (2,  -J53,  1).  4.     (3,  |,  i). 

5-    ■^  =  ^  =  ^^-  10-    8.7;  +  2/-26.s  +  6  =  0. 

14  1  ■      11  8  7     ■ 

9.    2x  +  5w-s  =  19.  12     ^•'  -  ^1  -  y  -  yi  =  g  -  '^1 

^  ^  C 

13.    a{x  -xi)  +  h(y  -  ?/i)  +  (•(£•  -  ,fi)  =  0. 

Art.  183.     Page  239. 

1.  x'^-y^-z-  =  o. 

2.  (a)  3x  +  2y  =  2. 

{b)  2x^  +  2y-^  +  6yz  +  6xz  +  5xy-2x-4ij-8s  +  i  =  0. 
(c)  6  x2  +  6  2/2  -  4  ^-^  +  15  x?/  -  18  X  -  18  y  +  12  =  0. 
(fZ)  12  j/2  4. 1.5  5^-2  +  12  a-y  -  8  a;  -  28  ?/  +12  =  0. 

Art.  186.     Pages  242-243. 

1.   X-  oz=::k,  X  +Sz  =  vi.  2.   k  =  7n  =  ±  \/2. 

„     4^  ,   Vl94„      ,    4  V194 

'•   3^+-T3-^  =  ''3^--l3-'  =  '"- 


INDEX 


(The  numbers  refer  to  the  pages.) 


Abscissa,  11. 

Addition  of  directed  segments,  angles, 

9. 
Adiabatic  exi^ansion,  192. 
Agnesi,  Donna  Maria,  172. 
Algebraic  functions,  41. 
Angle  which  one  segment  makes  with 
another,  22. 
which  one  line  makes  with  another, 

105. 
which  a  line  makes  with  a  plane, 

226. 
which  one   plane  makes   with   an- 
other, 218. 
Area  of  a  triangle,  26,  28. 

of  any  polygon,  30. 
Asymptotes,  81,  133. 
Asymptotic  cones,  243. 
Axes  of  coordinates,  11. 
of  ellipse,  62. 
of  hyperbola,  65. 
Axis  of  parabola,  123. 

of  pencil  of  planes,  219. 
Azimuth,  10. 

B 

Bisectors  of  angles,  106. 
Boyle's  law,  50. 


Cardioid,  178. 
Cartesian  coordinates,  10. 
Cassinian  ovals,  67. 
Catenary,  85. 


Circle,  55. 
Circular  cone,  211. 

sections,  242. 
Cissoid,  169. 
Classification  of  curves,  96. 

of  quadric  surfaces,  229. 
Clockwise,  9. 
Cofactors,  150,  217. 
Colatitude,  196. 
Common  chord,  163. 
Conchoid,  170. 
Cone,  236. 
Conicoids,  229. 

Conies  as  sections  of  a  cone,  211. 
Conic  sections,  110. 
Conjugate  axis,  62,  65. 

diameters,  123. 

hyperbolas,  134. 
Construction  of  a  surface,  228. 
Contour  lines,  136. 
Coordinate  axes,  11,  195. 

planes,  195. 
Coordinates,  cartesian,  11,  195. 

cylindrical,  197. 

of  point  of  contact,  115. 

polar,  13. 

rectangular,  11. 

spherical,  196. 
Cosine  curve,  43. 
Counterclockwise,  9. 
Cross-sections,  229. 
Cubic  curve,  83. 
Curves,  algebraic,  169. 

in  space,  245. 
Cusp,  176. 
Cycloid,  173. 
Cylinders,  210. 


263 


264 


INDEX 


D 

Damped  vibrations,  87. 
Descartes,  10. 
Determinant,  26. 

form  of  equation,  69,  216. 
Determination    of    functional    corre- 
spondence, 33. 
Diameter  of  conic,  123. 
Diodes,  170. 
Directed  segments,  8. 

angles,  8. 
Direction  cosines,  200. 
Director  circle,  114. 
Directrices  of  conies,  107. 
Directrix  of  a  parabola,  66. 
Discriminant,  150. 
Discussion  of  an  equation,  77. 
Distance  between  two  points,  20,  199. 

of  a  point  from  a  line,  104. 

of  a  point  from  a  plane,  217. 
Duplication  of  cube,  170. 

E 

Eccentricity,  of  ellipse,  62. 

of  hyperbola,  6-5. 
Ellipse,  60. 
Ellipsoid,  2.30. 

of  revolution,  210. 
Elliptic  paraboloid,  2-34. 
Empirical  equations,  182. 
Epicycloid,  177. 
Equations  of  a  line,  58,  221. 

of  first  degree,  98. 

of  a  plane,  213-216. 

of  second  degree,  107,  229. 

of  higher  degree,  169. 

of  tangents.  111. 
Equilateral  hyperbola,  135. 
Exponential  curve,  44. 


Foci  of  an  ellipse,  61. 

of  an  hyperbola,  64. 

of  a  cassinian  oval,  68. 
Eocus  of  a  parabola,  66. 
Folium  of  Descartes,  84. 


Four-cusped  hypocycloid,  177. 
Function,  33. 

algebraic,  41. 

inverse,  45. 

ti-anscendental,  41. 

G 

General  equation  of  second  degree, 

145. 
Graph  of  exponential  function,  44. 
Graphic  representation,  34. 
Graphs,  34. 
geometric  construction  of,  34,   42, 

44,  47. 
of  inverse  functions,  46. 
of  transcendental  functions,  41. 

H 

Harmonic  range,  131. 
Helix,  246. 
Hyperbola,  63. 
Hyperbolic  paraboloid,  235. 
Hyperboloid,  231,  233. 
Hypocycloid,  175. 


Intercept  form,  58,  215. 
Intercepts,  38. 
Intersecting  lines,  99. 

planes,  223. 
Inverse  functions,  45. 
Involute  of  a  circle,  180. 


Latus  rectum,  63,  66,  67. 

Law,  53. 

Lemniscate,  68. 

Length  of  a  segment,  20,  199. 

LimaQon,  172. 

Limiting  cases  of  conies,  143,  155. 

of  quadric  surfaces,  236-237. 
Line,  perpendicular  to  a  plane,  226. 

through  a  point,  101. 

through  two  points,  57. 


INDEX 


265 


Linear  e(|uations,  98, 

scale,  7. 
Lituus,  i)l. 
Locus  of  a  point,  53. 
Logaritbuiic  paper,  189. 

■spiral,  9L 

M 

Machines,  48. 
Major  axis,  62. 
Maximum,  o5. 
Midpoint  of  segment.  24. 
Minimum,  35. 
Minor  axis.  ()2. 
Monotone  function,  35. 
Multiple-valued  functions,  30. 

N 

Naperian  logarithms,  6. 
Nicomedes,  17  L 
Normal  form,  102,  213. 
Normal  to  a  curve,  1 19. 

0 

Oblate,  210. 

Oblique  axes,  11,  196. 

Ordinate,  11. 

Origin,  8. 

Orthogonal  sets  of  curves,  137. 


Pairs  of  planes,  237. 
Parabola,  66. 

cartesian  equation,  66. 

polar  equation,  71. 
Paraboloid,  elliptic,  231. 

hyperbolic,  235. 

of  revolution,  210. 
Parallel  segments,  23. 
Parameter,  73. 
Parametric  equations.  73. 
Pascal,  172. 

limagon,  172. 
Pencil  of  conies,  163. 

of  lines,  100. 


Pencil  of  planes,  219-220. 
Periodic  functions,  43. 
Perpendicular  segments,  23. 
Plane,  213. 

through  three  points,  215. 
Plotting,  34. 

Point  bisecting  a  segment,  24. 
dividing  a  segment  in  a  given  ratio, 
25. 
Polar  line,  128. 
coordinates,  13,  197. 
equation  of  circle,  69. 
equation  of  ellipse,  71. 
equation  of  hyperbola,  71. 
equation  of  line,  70. 
equation  of  parabola,  71. 
!  Poles  and  polars,  127. 
I  Position  of  a  ix)nit  in  a  plane,  10 
j  Profile,  l(i. 
Projectile,  180. 
Projecting  cylinders,  244. 

planes,  222. 
Projections  of  a  segment,  18,  202. 
Prolate.  210. 


Quadrant,  12. 
Quadratic  equation,  1 . 
Quadric  surfaces.  229. 

R 

Radical  axis,  163. 

Radius  vector,  13. 

Rectangular  coordinates,  11,  196. 
hyi^erbola,  73. 

Reduction  to  normal  form,  103. 

Reflection  properties,  120. 

Relation  between  rectangular  coordi- 
nates and  polar  coordinates,  14. 

Removal  of  term  in  :nj,  95. 

Rotation  of  axes,  92. 

Ruled  surfaces,  238. 

Rulings  on  hyperboloids,  239. 


Semicubical  parabola,  173. 
Simultaneous  linear  equations,  99. 


266 


INDEX 


Sine  curve,  42. 
Single-valued  functions,  36. 
Slope  of  segment,  18. 

form  of  equation  of  a  line,  58. 
Sphere,  208. 

Splierical  coordinates,  196. 
Spiral  of  Archimedes,  40. 
Steam  pressure  gauge,  49. 
Strophoid,  97. 
Subnormal,  119. 
Subtangent,  119. 
Surface  of  revolution,  209. 
Symmetry,  37,  77,  228. 
System  of  concentric  hyperbolas,  135. 

of  circles,  163. 

of  confocal  conies,  109. 


Tangent  plane,  240. 
Tangent  to  a  curve.  111. 

to  a  circle,  111. 

to  an  ellipse,  112. 


Tangent  to  an  hyperbola,  112. 

to  a  parabola,  112. 
Temperature  a  function  of  time,  50. 
Three-cusped  hypocycloid,  180. 
Transcendental  curves,  173. 

functions,  41. 
Transformation  of  coordinates,  91. 

from  cartesian  to  polar,  14. 
Translation  of  axes,  92. 
Transverse  axis,  62,  65. 
Trochoid,  180. 
Trigonometric  formulas,  1. 
Trisection  of  an  angle,  171,  172. 
Trisectrix  of  Maclaurin,  173. 


Vertex  of  parabola,  66 

Vertices  of  ellipse,  62. 

of  hyperbola,  65. 

W 

Witch  of  Agnesi,  171. 


Date  Due 

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